[{"publication_identifier":{"eisbn":["9781611978971"]},"oa":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2409.20314","open_access":"1"}],"publication":"Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms","author":[{"first_name":"Pavel","id":"b25f2ab2-1fed-11ee-8599-fe02d211784f","last_name":"Arkhipov","full_name":"Arkhipov, Pavel"},{"first_name":"Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov"}],"quality_controlled":"1","abstract":[{"lang":"eng","text":"We consider several problems related to packing forests in graphs. The first one is to find k edge-disjoint forests in a directed graph G of maximal size such that the indegree of each vertex in these forests is at most k. We describe a min-max characterization for this problem and show that it can be solved in almost linear time for fixed k, extending the algorithm of [Gabow, 1995]. Specifically, the complexity is O(kδm log n), where n, m are the number of vertices and edges in G respectively, and δ = max{1, k − kG}, where kG is the edge connectivity of the graph. Using our solution to this problem, we improve complexities for two existing applications:(1) k-forest problem: find k forests in an undirected graph G maximizing the number of edges in their union. We show how to solve this problem in O(k3 min{kn, m} log2 n + k · MAXFLOW(m, m) log n) time, breaking the Ok(n3/2) complexity barrier of previously known approaches.(2) Directed edge-connectivity augmentation problem: find a smallest set of directed edges whose addition to the given directed graph makes it strongly k-connected. We improve the deterministic complexity for this problem from O(kδ(m + δn) log n) [Gabow, STOC 1994] to O(kδm log n). A similar approach with the same complexity also works for the undirected version of the problem."}],"publication_status":"published","department":[{"_id":"VlKo"}],"external_id":{"arxiv":["2409.20314"]},"date_created":"2026-02-05T10:51:34Z","corr_author":"1","language":[{"iso":"eng"}],"status":"public","day":"07","publisher":"Society for Industrial and Applied Mathematics","type":"conference","date_published":"2026-01-07T00:00:00Z","year":"2026","oa_version":"Preprint","date_updated":"2026-02-16T09:18:33Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"4023-4042","_id":"21140","title":"Faster algorithms for packing forests in graphs and related problems","month":"01","article_processing_charge":"No","conference":{"name":"SODA: Symposium on Discrete Algorithms","start_date":"2026-01-11","location":"Vancouver, Canada","end_date":"2026-01-14"},"citation":{"mla":"Arkhipov, Pavel, and Vladimir Kolmogorov. “Faster Algorithms for Packing Forests in Graphs and Related Problems.” Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2026, pp. 4023–42, doi:10.1137/1.9781611978971.148.","short":"P. Arkhipov, V. Kolmogorov, in:, Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2026, pp. 4023–4042.","ieee":"P. Arkhipov and V. Kolmogorov, “Faster algorithms for packing forests in graphs and related problems,” in Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, Vancouver, Canada, 2026, pp. 4023–4042.","apa":"Arkhipov, P., & Kolmogorov, V. (2026). Faster algorithms for packing forests in graphs and related problems. In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 4023–4042). Vancouver, Canada: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611978971.148","ama":"Arkhipov P, Kolmogorov V. Faster algorithms for packing forests in graphs and related problems. In: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics; 2026:4023-4042. doi:10.1137/1.9781611978971.148","ista":"Arkhipov P, Kolmogorov V. 2026. Faster algorithms for packing forests in graphs and related problems. Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 4023–4042.","chicago":"Arkhipov, Pavel, and Vladimir Kolmogorov. “Faster Algorithms for Packing Forests in Graphs and Related Problems.” In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms, 4023–42. Society for Industrial and Applied Mathematics, 2026. https://doi.org/10.1137/1.9781611978971.148."},"arxiv":1,"doi":"10.1137/1.9781611978971.148","OA_place":"repository","OA_type":"green"},{"page":"18","author":[{"first_name":"Ivan","id":"ca3c9187-9a72-11ee-a009-8af825d896b0","last_name":"Sergeev","full_name":"Sergeev, Ivan","orcid":"0009-0004-9145-8785"},{"last_name":"Dvorak","full_name":"Dvorak, Martin","first_name":"Martin","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","orcid":"0000-0001-5293-214X"},{"first_name":"Cameron","last_name":"Rampell","full_name":"Rampell, Cameron"},{"last_name":"Sandey","full_name":"Sandey, Mark","first_name":"Mark"},{"first_name":"Pietro","last_name":"Monticone","full_name":"Monticone, Pietro"}],"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","publication":"arXiv","oa_version":"Preprint","year":"2026","date_updated":"2026-03-09T15:14:18Z","date_published":"2026-01-03T00:00:00Z","type":"preprint","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2601.01255","open_access":"1"}],"day":"03","oa":1,"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","OA_place":"repository","language":[{"iso":"eng"}],"doi":"10.48550/arXiv.2601.01255","corr_author":"1","department":[{"_id":"GradSch"},{"_id":"VlKo"}],"external_id":{"arxiv":["2601.01255"]},"date_created":"2026-03-04T12:09:26Z","publication_status":"submitted","arxiv":1,"citation":{"apa":"Sergeev, I., Dvorak, M., Rampell, C., Sandey, M., & Monticone, P. (n.d.). A blueprint for the formalization of Seymour’s matroid decomposition theorem. arXiv. https://doi.org/10.48550/arXiv.2601.01255","ama":"Sergeev I, Dvorak M, Rampell C, Sandey M, Monticone P. A blueprint for the formalization of Seymour’s matroid decomposition theorem. arXiv. doi:10.48550/arXiv.2601.01255","ista":"Sergeev I, Dvorak M, Rampell C, Sandey M, Monticone P. A blueprint for the formalization of Seymour’s matroid decomposition theorem. arXiv, 10.48550/arXiv.2601.01255.","chicago":"Sergeev, Ivan, Martin Dvorak, Cameron Rampell, Mark Sandey, and Pietro Monticone. “A Blueprint for the Formalization of Seymour’s Matroid Decomposition Theorem.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2601.01255.","short":"I. Sergeev, M. Dvorak, C. Rampell, M. Sandey, P. Monticone, ArXiv (n.d.).","mla":"Sergeev, Ivan, et al. “A Blueprint for the Formalization of Seymour’s Matroid Decomposition Theorem.” ArXiv, doi:10.48550/arXiv.2601.01255.","ieee":"I. Sergeev, M. Dvorak, C. Rampell, M. Sandey, and P. Monticone, “A blueprint for the formalization of Seymour’s matroid decomposition theorem,” arXiv. ."},"related_material":{"link":[{"relation":"supplementary_material","url":"https://ivan-sergeyev.github.io/seymour/blueprint.pdf"}]},"abstract":[{"text":"This document is a blueprint for the formalization in Lean of the structural theory of regular matroids underlying Seymour's decomposition theorem. We present a modular account of regularity via totally unimodular representations, show that regularity is preserved under 1-, 2-, and 3-sums, and establish regularity for several special classes of matroids, including graphic, cographic, and the matroid R10. The blueprint records the logical structure of the proof, the precise dependencies between results, and their correspondence with Lean declarations. It is intended both as a guide for the ongoing formalization effort and as a human-readable reference for the organization of the proof.","lang":"eng"}],"article_processing_charge":"No","title":"A blueprint for the formalization of Seymour's matroid decomposition theorem","month":"01","_id":"21400"},{"related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"13120"},{"id":"21398","relation":"part_of_dissertation","status":"public"},{"status":"public","relation":"part_of_dissertation","id":"20071"}],"link":[{"description":"Full version of all definitions, statements, and proofs for Chapter 3.1 (Linear duality)","url":"https://github.com/madvorak/duality/tree/v3.5.0","relation":"software"},{"description":"Full version of all definitions, statements, and proofs for Chapter 3.2 (Valued Constraint Satisfaction Problems)","relation":"software","url":"https://github.com/madvorak/vcsp/tree/v8.2.0"},{"description":"Full version of all definitions, statements, and proofs for Chapter 4 (Seymour project)","url":"https://github.com/Ivan-Sergeyev/seymour/tree/v1.2.0","relation":"software"},{"relation":"software","url":"https://github.com/madvorak/chomsky/tree/v1.2.0","description":"Full version of all definitions, statements, and proofs for Chapter 5 (Theory of grammars)"},{"url":"https://github.com/madvorak/grammars","relation":"software","description":"Old version (Lean 3) of the project about grammars"},{"url":"https://github.com/madvorak/preliminaries/blob/main/Preliminaries.lean","relation":"software","description":"Demonstration of (minimal) requirements for selected algebraic classes used in my Ph.D. thesis"}]},"citation":{"chicago":"Dvorak, Martin. “Pursuit of Truth and Beauty in Lean 4 : Formally Verified Theory of Grammars, Optimization, Matroids.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-21393.","ista":"Dvorak M. 2026. Pursuit of truth and beauty in Lean 4 : Formally verified theory of grammars, optimization, matroids. Institute of Science and Technology Austria.","ama":"Dvorak M. Pursuit of truth and beauty in Lean 4 : Formally verified theory of grammars, optimization, matroids. 2026. doi:10.15479/AT-ISTA-21393","apa":"Dvorak, M. (2026). Pursuit of truth and beauty in Lean 4 : Formally verified theory of grammars, optimization, matroids. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-21393","ieee":"M. Dvorak, “Pursuit of truth and beauty in Lean 4 : Formally verified theory of grammars, optimization, matroids,” Institute of Science and Technology Austria, 2026.","short":"M. Dvorak, Pursuit of Truth and Beauty in Lean 4 : Formally Verified Theory of Grammars, Optimization, Matroids, Institute of Science and Technology Austria, 2026.","mla":"Dvorak, Martin. Pursuit of Truth and Beauty in Lean 4 : Formally Verified Theory of Grammars, Optimization, Matroids. Institute of Science and Technology Austria, 2026, doi:10.15479/AT-ISTA-21393."},"article_processing_charge":"No","month":"03","title":"Pursuit of truth and beauty in Lean 4 : Formally verified theory of grammars, optimization, matroids","_id":"21393","alternative_title":["ISTA Thesis"],"OA_place":"repository","doi":"10.15479/AT-ISTA-21393","ddc":["511","000"],"day":"04","type":"dissertation","publisher":"Institute of Science and Technology Austria","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","page":"160","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","date_updated":"2026-03-27T12:37:00Z","oa_version":"Published Version","year":"2026","has_accepted_license":"1","date_published":"2026-03-04T00:00:00Z","supervisor":[{"id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir","full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov"},{"last_name":"Blanchette","full_name":"Blanchette, Jasmin","first_name":"Jasmin"}],"abstract":[{"text":"This thesis documents a voyage towards truth and beauty via formal verification of theorems. To this end, we develop libraries in Lean 4 that present definitions and results from diverse areas of MathematiCS (i.e., Mathematics and Computer Science). The aim is to create code that is understandable, believable, useful, and elegant. The code should stand for itself as much as possible without a need for documentation; however, this text redundantly documents our code artifacts and provides additional context that isn’t present in the code. This thesis is written for readers who know Lean 4 but are not familiar with any of the topics presented. We manifest truth and beauty in three formalized areas of MathematiCS.\r\n\r\nWe formalize general grammars in Lean 4 and use grammars to show closure of the class of type-0 languages under four operations; union, reversal, concatenation, and the Kleene star.\r\n\r\nOur second stop is the theory of optimization. Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take “infinite values”. Additionally, we develop the basics of the theory of optimization in terms of the framework called General-Valued Constraint Satisfaction Problems, and we prove that, if a Rational-Valued Constraint Satisfaction Problem template has symmetric fractional polymorphisms of all arities, then its basic LP relaxation is tight.\r\n\r\nOur third stop is matroid theory. Seymour’s decomposition theorem is a hallmark result in matroid theory, presenting a structural characterization of the class of regular matroids. We aim to formally verify Seymour’s theorem in Lean 4. First, we build a library for working with totally unimodular matrices. We define binary matroids and their standard representations, and we prove that they form a matroid in the sense how Mathlib defines matroids. We define regular matroids to be matroids for which there exists a full representation rational matrix that is totally unimodular, and we prove that all regular matroids are binary. We define 1-sum, 2-sum, and 3 sum of binary matroids as specific ways to compose their standard representation matrices. We prove that the 1-sum, the 2-sum, and the 3-sum of regular matroids are a regular matroid, which concludes the composition direction of the Seymour’s theorem. The (more difficult) decomposition direction remains unproved.\r\n\r\nIn the pursuit of truth, we focus on identifying the trusted code in each project and presenting it faithfully. We emphasize the readability and believability of definitions rather than choosing definitions that are easier to work with. In search for beauty, we focus on the philosophical framework of Roger Scruton, who emphasizes that beauty is not a mere decoration but, most importantly, beauty is the means for shaping our place in the world and a source of redemption, where it can be viewed as a substitute for religion.","lang":"eng"}],"language":[{"iso":"eng"}],"corr_author":"1","date_created":"2026-03-04T09:26:46Z","department":[{"_id":"GradSch"},{"_id":"VlKo"}],"publication_status":"published","degree_awarded":"PhD","publication_identifier":{"issn":["2663-337X"],"isbn":["978-3-99078-074-9"]},"oa":1,"file":[{"access_level":"open_access","relation":"main_file","success":1,"checksum":"cface6dc18152680962b5361575f6e4f","content_type":"application/pdf","date_updated":"2026-03-04T08:56:15Z","file_name":"2026_Dvorak_Martin_Thesis.pdf","creator":"mdvorak","file_id":"21394","date_created":"2026-03-04T08:56:15Z","file_size":1771231},{"relation":"source_file","access_level":"closed","date_updated":"2026-03-04T09:03:37Z","content_type":"application/vnd.openxmlformats-officedocument.wordprocessingml.document","checksum":"290ddfacfb7e07fb07e6f0b334e67c90","file_size":864585,"file_id":"21395","date_created":"2026-03-04T09:03:37Z","file_name":"2026_Dvorak_Martin_Thesis.docx","creator":"mdvorak"}],"file_date_updated":"2026-03-04T09:03:37Z","author":[{"orcid":"0000-0001-5293-214X","first_name":"Martin","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","last_name":"Dvorak","full_name":"Dvorak, Martin"}]},{"publication":"ACM Transactions on Algorithms","author":[{"first_name":"David G.","full_name":"Harris, David G.","last_name":"Harris"},{"id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir","last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir"}],"scopus_import":"1","ec_funded":1,"intvolume":" 21","isi":1,"acknowledgement":"We thank Heng Guo for helpful explanations of algorithms for sampling connected subgraphs and matchings, and Maksym Serbyn for bringing to our attention the WL algorithm and its use in physics.\r\nThis is an extended version, which includes work under the same name from ICALP 2023, as well as the earlier work [22] appearing in COLT 2018.\r\nV. Kolmogorov was supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160","publication_identifier":{"eissn":["1549-6333"],"issn":["1549-6325"]},"oa":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2007.10824","open_access":"1"}],"publication_status":"published","external_id":{"arxiv":["2007.10824"],"isi":["001399998600008"]},"department":[{"_id":"VlKo"}],"date_created":"2025-01-19T23:01:52Z","corr_author":"1","language":[{"iso":"eng"}],"volume":21,"quality_controlled":"1","abstract":[{"text":"A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We consider sampling problems which come from Gibbs distributions, which are families of probability distributions over a discrete space Ω with probability mass function of the form μ^Ω_β(ω) ∝ e^{β H(ω)} for β in an interval [β_min, β_max] and H(ω) ∈ {0} ∪ [1, n]. Two important parameters are the partition function, which is the normalization factor Z(β) = ∑_{ω ∈ Ω} e^{β H(ω)}, and the vector of pre-image counts c_x=|H^-1(x)|.\r\nWe develop black-box sampling algorithms to estimate the counts roughly Õ(n²/ε²) samples for integer-valued distributions and Õ(q/ε²) samples for general distributions, where q = (log Z(β_max))/Z(β_min) (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we estimate all values of the partition function using Õ(n²/ε²) samples for integer-valued distributions and Õ(q/ε²) samples for general distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate Z(β_max) and which uses a slightly larger amount of samples. We show matching lower bounds, demonstrating this complexity is optimal as a function of n and q up to logarithmic terms.","lang":"eng"}],"article_number":"3","date_published":"2025-01-01T00:00:00Z","project":[{"_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160","name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7"}],"oa_version":"Preprint","year":"2025","date_updated":"2025-07-10T11:50:44Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","publisher":"Association for Computing Machinery","day":"01","type":"journal_article","issue":"1","arxiv":1,"OA_place":"repository","doi":"10.1145/3685676","article_type":"original","OA_type":"green","_id":"18855","title":"Parameter estimation for Gibbs distributions","month":"01","article_processing_charge":"No","citation":{"ista":"Harris DG, Kolmogorov V. 2025. Parameter estimation for Gibbs distributions. ACM Transactions on Algorithms. 21(1), 3.","chicago":"Harris, David G., and Vladimir Kolmogorov. “Parameter Estimation for Gibbs Distributions.” ACM Transactions on Algorithms. Association for Computing Machinery, 2025. https://doi.org/10.1145/3685676.","apa":"Harris, D. G., & Kolmogorov, V. (2025). Parameter estimation for Gibbs distributions. ACM Transactions on Algorithms. Association for Computing Machinery. https://doi.org/10.1145/3685676","ama":"Harris DG, Kolmogorov V. Parameter estimation for Gibbs distributions. ACM Transactions on Algorithms. 2025;21(1). doi:10.1145/3685676","ieee":"D. G. Harris and V. Kolmogorov, “Parameter estimation for Gibbs distributions,” ACM Transactions on Algorithms, vol. 21, no. 1. Association for Computing Machinery, 2025.","mla":"Harris, David G., and Vladimir Kolmogorov. “Parameter Estimation for Gibbs Distributions.” ACM Transactions on Algorithms, vol. 21, no. 1, 3, Association for Computing Machinery, 2025, doi:10.1145/3685676.","short":"D.G. Harris, V. Kolmogorov, ACM Transactions on Algorithms 21 (2025)."},"related_material":{"record":[{"id":"14084","status":"public","relation":"earlier_version"}]}},{"volume":209,"quality_controlled":"1","abstract":[{"lang":"eng","text":"Given a fixed finite metric space (V,μ), the {\\em minimum 0-extension problem}, denoted as 0-Ext[μ], is equivalent to the following optimization problem: minimize function of the form minx∈Vn∑ifi(xi)+∑ijcijμ(xi,xj) where cij,cvi are given nonnegative costs and fi:V→R are functions given by fi(xi)=∑v∈Vcviμ(xi,v). The computational complexity of 0-Ext[μ] has been recently established by Karzanov and by Hirai: if metric μ is {\\em orientable modular} then 0-Ext[μ] can be solved in polynomial time, otherwise 0-Ext[μ] is NP-hard. To prove the tractability part, Hirai developed a theory of discrete convex functions on orientable modular graphs generalizing several known classes of functions in discrete convex analysis, such as L♮-convex functions. We consider a more general version of the problem in which unary functions fi(xi) can additionally have terms of the form cuv;iμ(xi,{u,v}) for {u,v}∈F, where set F⊆(V2) is fixed. We extend the complexity classification above by providing an explicit condition on (μ,F) for the problem to be tractable. In order to prove the tractability part, we generalize Hirai's theory and define a larger class of discrete convex functions. It covers, in particular, another well-known class of functions, namely submodular functions on an integer lattice. Finally, we improve the complexity of Hirai's algorithm for solving 0-Ext on orientable modular graphs.\r\n"}],"publication_status":"published","external_id":{"isi":["001176563300001"],"arxiv":["2109.10203"]},"department":[{"_id":"GradSch"},{"_id":"VlKo"}],"date_created":"2021-09-27T10:48:23Z","corr_author":"1","language":[{"iso":"eng"}],"acknowledgement":"We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. Open access funding provided by Institute of Science and Technology (IST Austria).","file":[{"relation":"main_file","access_level":"open_access","success":1,"date_updated":"2025-04-16T09:36:08Z","content_type":"application/pdf","checksum":"25d9bd490719b45eca84f4d93a06c69f","file_size":839510,"date_created":"2025-04-16T09:36:08Z","file_id":"19578","creator":"dernst","file_name":"2025_MathProgramming_Dvorak.pdf"}],"oa":1,"publication_identifier":{"eissn":["1436-4646"],"issn":["0025-5610"]},"scopus_import":"1","file_date_updated":"2025-04-16T09:36:08Z","keyword":["minimum 0-extension problem","metric labeling problem","discrete metric spaces","metric extensions","computational complexity","valued constraint satisfaction problems","discrete convex analysis","L-convex functions"],"author":[{"orcid":"0000-0001-5293-214X","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","first_name":"Martin","full_name":"Dvorak, Martin","last_name":"Dvorak"},{"full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"}],"publication":"Mathematical Programming","intvolume":" 209","isi":1,"_id":"10045","title":"Generalized minimum 0-extension problem and discrete convexity","month":"01","article_processing_charge":"Yes (via OA deal)","citation":{"ieee":"M. Dvorak and V. Kolmogorov, “Generalized minimum 0-extension problem and discrete convexity,” Mathematical Programming, vol. 209. Springer Nature, pp. 279–322, 2025.","mla":"Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension Problem and Discrete Convexity.” Mathematical Programming, vol. 209, Springer Nature, 2025, pp. 279–322, doi:10.1007/s10107-024-02064-5.","short":"M. Dvorak, V. Kolmogorov, Mathematical Programming 209 (2025) 279–322.","chicago":"Dvorak, Martin, and Vladimir Kolmogorov. “Generalized Minimum 0-Extension Problem and Discrete Convexity.” Mathematical Programming. Springer Nature, 2025. https://doi.org/10.1007/s10107-024-02064-5.","ista":"Dvorak M, Kolmogorov V. 2025. Generalized minimum 0-extension problem and discrete convexity. Mathematical Programming. 209, 279–322.","ama":"Dvorak M, Kolmogorov V. Generalized minimum 0-extension problem and discrete convexity. Mathematical Programming. 2025;209:279-322. doi:10.1007/s10107-024-02064-5","apa":"Dvorak, M., & Kolmogorov, V. (2025). Generalized minimum 0-extension problem and discrete convexity. Mathematical Programming. Springer Nature. https://doi.org/10.1007/s10107-024-02064-5"},"arxiv":1,"ddc":["004"],"doi":"10.1007/s10107-024-02064-5","OA_place":"publisher","article_type":"original","OA_type":"hybrid","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"type":"journal_article","publisher":"Springer Nature","day":"01","has_accepted_license":"1","date_published":"2025-01-01T00:00:00Z","oa_version":"Published Version","year":"2025","date_updated":"2025-05-19T13:52:10Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"279-322"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"1-22","has_accepted_license":"1","date_published":"2025-10-01T00:00:00Z","date_updated":"2026-01-21T09:46:26Z","oa_version":"Published Version","year":"2025","issue":"4","day":"01","publisher":"Association for Computing Machinery","type":"journal_article","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"OA_type":"hybrid","article_type":"original","doi":"10.1145/3748723","OA_place":"publisher","ddc":["510"],"arxiv":1,"article_processing_charge":"Yes (via OA deal)","related_material":{"record":[{"relation":"shorter_version","status":"public","id":"17236"}]},"PlanS_conform":"1","citation":{"mla":"Kolmogorov, Vladimir. “A Simpler and Parallelizable O(√log n)-Approximation Algorithm for SPARSEST CUT.” ACM Transactions on Algorithms, vol. 21, no. 4, Association for Computing Machinery, 2025, pp. 1–22, doi:10.1145/3748723.","short":"V. Kolmogorov, ACM Transactions on Algorithms 21 (2025) 1–22.","ieee":"V. Kolmogorov, “A simpler and parallelizable O(√log n)-approximation algorithm for SPARSEST CUT,” ACM Transactions on Algorithms, vol. 21, no. 4. Association for Computing Machinery, pp. 1–22, 2025.","apa":"Kolmogorov, V. (2025). A simpler and parallelizable O(√log n)-approximation algorithm for SPARSEST CUT. ACM Transactions on Algorithms. Association for Computing Machinery. https://doi.org/10.1145/3748723","ama":"Kolmogorov V. A simpler and parallelizable O(√log n)-approximation algorithm for SPARSEST CUT. ACM Transactions on Algorithms. 2025;21(4):1-22. doi:10.1145/3748723","ista":"Kolmogorov V. 2025. A simpler and parallelizable O(√log n)-approximation algorithm for SPARSEST CUT. ACM Transactions on Algorithms. 21(4), 1–22.","chicago":"Kolmogorov, Vladimir. “A Simpler and Parallelizable O(√log n)-Approximation Algorithm for SPARSEST CUT.” ACM Transactions on Algorithms. Association for Computing Machinery, 2025. https://doi.org/10.1145/3748723."},"_id":"21007","month":"10","title":"A simpler and parallelizable O(√log n)-approximation algorithm for SPARSEST CUT","scopus_import":"1","author":[{"last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"}],"publication":"ACM Transactions on Algorithms","file_date_updated":"2026-01-21T09:38:09Z","intvolume":" 21","file":[{"checksum":"4a80fdb1e3711b9a2768d2bb8f6d3b4e","content_type":"application/pdf","date_updated":"2026-01-21T09:38:09Z","file_name":"2025_ACMToA_Kolmogorov.pdf","creator":"dernst","date_created":"2026-01-21T09:38:09Z","file_id":"21031","file_size":2208302,"access_level":"open_access","relation":"main_file","success":1}],"publication_identifier":{"issn":["1549-6325"],"eissn":["1549-6333"]},"oa":1,"date_created":"2026-01-20T10:04:02Z","department":[{"_id":"VlKo"}],"external_id":{"arxiv":["2307.00115"]},"language":[{"iso":"eng"}],"corr_author":"1","publication_status":"published","abstract":[{"text":"Currently, the best known tradeoff between approximation ratio and complexity for the Sparsest Cut problem is achieved by the algorithm in [Sherman, FOCS 2009]: it computes O(√(log n)/ε)-approximation using O(nε logO(1) n) maxflows for any ε∈[Θ(1/log n),Θ(1)]. It works by solving the SDP relaxation of [Arora-Rao-Vazirani, STOC 2004] using the Multiplicative Weights Update algorithm (MW) of [Arora-Kale, JACM 2016]. To implement one MW step, Sherman approximately solves a multicommodity flow problem using another application of MW. Nested MW steps are solved via a certain \"chaining\" algorithm that combines results of multiple calls to the maxflow algorithm. We present an alternative approach that avoids solving the multicommodity flow problem and instead computes \"violating paths\". This simplifies Sherman's algorithm by removing a need for a nested application of MW, and also allows parallelization: we show how to compute O(√(log n)/ε)-approximation via O(logO(1) n) maxflows using O(nε) processors. We also revisit Sherman's chaining algorithm, and present a simpler version together with a new analysis.","lang":"eng"}],"volume":21,"quality_controlled":"1"},{"publication":"Theory of Computing","file_date_updated":"2026-02-10T09:54:28Z","author":[{"full_name":"Harris, David G.","last_name":"Harris","first_name":"David G."},{"first_name":"Fotios","full_name":"Iliopoulos, Fotios","last_name":"Iliopoulos"},{"first_name":"Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir"}],"intvolume":" 21","ec_funded":1,"file":[{"success":1,"relation":"main_file","access_level":"open_access","file_size":509346,"date_created":"2026-02-10T09:54:28Z","file_id":"21209","file_name":"2025_TheoryComputing_Harris.pdf","creator":"dernst","date_updated":"2026-02-10T09:54:28Z","checksum":"5a9f7cfccac6046fe75a14a4059eed04","content_type":"application/pdf"}],"acknowledgement":"This material is based on work directly supported by the IAS Fund for Math and indirectly supported by the National Science Foundation Grant No. CCF-1900460. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work is also supported by the National Science Foundation Grant No. CCF-1815328. Supported by the European Research Council under the European Union’s Seventh Framework Programme\r\n(FP7/2007-2013)/ERC grant agreement no 616160.","oa":1,"publication_identifier":{"eissn":["1557-2862"]},"publication_status":"published","date_created":"2026-02-05T12:04:58Z","external_id":{"arxiv":["2008.05569"]},"department":[{"_id":"VlKo"}],"language":[{"iso":"eng"}],"corr_author":"1","volume":21,"quality_controlled":"1","abstract":[{"lang":"eng","text":"The Lovász Local Lemma (LLL) is a powerful tool in probabilistic\r\ncombinatorics which can be used to establish the existence of objects with certain\r\nproperties. The breakthrough paper by Moser & Tardos (STOC’09 and JACM 2010)\r\nand follow-up work revealed that the LLL has intimate connections with a class of\r\nstochastic local search algorithms for finding such desirable objects.\r\nBesides conditions for convergence, many other natural questions can be asked\r\nabout algorithms; for instance, “are they parallelizable?”, “how many solutions can\r\nthey output?”, “what is the expected ‘weight’ of a solution?”. These questions and\r\nmore have been answered for a class of LLL-inspired algorithms called commutative. In\r\nthis paper we introduce a new, very natural and more general notion of commutativity\r\n(essentially matrix commutativity) which allows us to show a number of new refined\r\nproperties of LLL-inspired local search algorithms with significantly simpler proofs."}],"project":[{"_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160","name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7"}],"date_published":"2025-09-08T00:00:00Z","has_accepted_license":"1","date_updated":"2026-02-10T10:00:00Z","oa_version":"Published Version","year":"2025","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"1 - 34","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"issue":"5","day":"08","type":"journal_article","publisher":"University of Chicago Press","ddc":["510"],"arxiv":1,"article_type":"original","OA_type":"diamond","OA_place":"publisher","doi":"10.4086/toc.2025.v021a005","_id":"21143","month":"09","title":"A new notion of commutativity for the algorithmic Lovász Local Lemma","article_processing_charge":"No","PlanS_conform":"1","related_material":{"record":[{"id":"10072","relation":"earlier_version","status":"public"}]},"citation":{"ama":"Harris DG, Iliopoulos F, Kolmogorov V. A new notion of commutativity for the algorithmic Lovász Local Lemma. Theory of Computing. 2025;21(5):1-34. doi:10.4086/toc.2025.v021a005","apa":"Harris, D. G., Iliopoulos, F., & Kolmogorov, V. (2025). A new notion of commutativity for the algorithmic Lovász Local Lemma. Theory of Computing. University of Chicago Press. https://doi.org/10.4086/toc.2025.v021a005","chicago":"Harris, David G., Fotios Iliopoulos, and Vladimir Kolmogorov. “A New Notion of Commutativity for the Algorithmic Lovász Local Lemma.” Theory of Computing. University of Chicago Press, 2025. https://doi.org/10.4086/toc.2025.v021a005.","ista":"Harris DG, Iliopoulos F, Kolmogorov V. 2025. A new notion of commutativity for the algorithmic Lovász Local Lemma. Theory of Computing. 21(5), 1–34.","short":"D.G. Harris, F. Iliopoulos, V. Kolmogorov, Theory of Computing 21 (2025) 1–34.","mla":"Harris, David G., et al. “A New Notion of Commutativity for the Algorithmic Lovász Local Lemma.” Theory of Computing, vol. 21, no. 5, University of Chicago Press, 2025, pp. 1–34, doi:10.4086/toc.2025.v021a005.","ieee":"D. G. Harris, F. Iliopoulos, and V. Kolmogorov, “A new notion of commutativity for the algorithmic Lovász Local Lemma,” Theory of Computing, vol. 21, no. 5. University of Chicago Press, pp. 1–34, 2025."}},{"day":"01","type":"journal_article","publisher":"Society for Industrial and Applied Mathematics","issue":"3","status":"public","page":"1630-1654","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2025","oa_version":"Preprint","date_updated":"2026-02-16T11:52:23Z","date_published":"2025-09-01T00:00:00Z","citation":{"ieee":"V. Kolmogorov, S. Naldi, and J. Zapata, “Certifying solutions of degenerate semidefinite programs,” SIAM Journal on Optimization, vol. 35, no. 3. Society for Industrial and Applied Mathematics, pp. 1630–1654, 2025.","mla":"Kolmogorov, Vladimir, et al. “Certifying Solutions of Degenerate Semidefinite Programs.” SIAM Journal on Optimization, vol. 35, no. 3, Society for Industrial and Applied Mathematics, 2025, pp. 1630–54, doi:10.1137/24m1664691.","short":"V. Kolmogorov, S. Naldi, J. Zapata, SIAM Journal on Optimization 35 (2025) 1630–1654.","chicago":"Kolmogorov, Vladimir, Simone Naldi, and Jeferson Zapata. “Certifying Solutions of Degenerate Semidefinite Programs.” SIAM Journal on Optimization. Society for Industrial and Applied Mathematics, 2025. https://doi.org/10.1137/24m1664691.","ista":"Kolmogorov V, Naldi S, Zapata J. 2025. Certifying solutions of degenerate semidefinite programs. SIAM Journal on Optimization. 35(3), 1630–1654.","ama":"Kolmogorov V, Naldi S, Zapata J. Certifying solutions of degenerate semidefinite programs. SIAM Journal on Optimization. 2025;35(3):1630-1654. doi:10.1137/24m1664691","apa":"Kolmogorov, V., Naldi, S., & Zapata, J. (2025). Certifying solutions of degenerate semidefinite programs. SIAM Journal on Optimization. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/24m1664691"},"article_processing_charge":"No","title":"Certifying solutions of degenerate semidefinite programs","month":"09","_id":"21144","doi":"10.1137/24m1664691","OA_place":"repository","OA_type":"green","article_type":"original","arxiv":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2405.13625","open_access":"1"}],"oa":1,"publication_identifier":{"issn":["1052-6234"],"eissn":["1095-7189"]},"intvolume":" 35","author":[{"last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"},{"first_name":"Simone","full_name":"Naldi, Simone","last_name":"Naldi"},{"id":"00223538-AF8F-11E9-A4C7-F729E6697425","first_name":"Jeferson","last_name":"Zapata","full_name":"Zapata, Jeferson"}],"publication":"SIAM Journal on Optimization","abstract":[{"lang":"eng","text":"This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMIs). Since in some SDP instances all feasible solutions have irrational entries, numerical solvers that work with rational numbers can only find an approximate solution. We study the following question: Is it possible to certify feasibility of a given SDP using an approximate solution that is sufficiently close to some exact solution? Existing approaches make the assumption that there exist rational feasible solutions (and use techniques such as rounding and lattice reduction algorithms). We propose an alternative approach that does not need this assumption. More specifically, we show how to construct a system of polynomial equations whose set of real solutions is guaranteed to have an isolated correct solution (assuming that the target exact solution is maximum-rank). This allows, in particular, for us to use algorithms from real algebraic geometry for solving systems of polynomial equations, yielding a hybrid (or symbolic-numerical) method for SDPs. We experimentally compare it with a pure symbolic method in [D. Henrion, S. Naldi, and M. Safey El Din, SIAM J. Optim., 26 (2016), pp. 2512–2539]; the hybrid method was able to certify feasibility of many SDP instances on which the aforementioned paper failed. Our approach may have further applications, such as refining an approximate solution using methods of numerical algebraic geometry for systems of polynomial equations."}],"quality_controlled":"1","volume":35,"language":[{"iso":"eng"}],"department":[{"_id":"VlKo"},{"_id":"GradSch"}],"external_id":{"arxiv":["2405.13625"]},"date_created":"2026-02-05T13:33:05Z","publication_status":"published"},{"abstract":[{"text":"We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.","lang":"eng"}],"article_processing_charge":"No","citation":{"short":"M. Bolan, J. Breitner, J. Brox, N. Carlini, M. Carneiro, F. van Doorn, M. Dvorak, A. Goens, A. Hill, H. Husum, H.I. Mejia, Z.A. Kocsis, B.L. Floch, A. Bar-on, L. Luccioli, D. McNeil, A. Meiburg, P. Monticone, P.P. Nielsen, E.O. Osazuwa, G. Paolini, M. Petracci, B. Reinke, D. Renshaw, M. Rossel, C. Roux, J. Scanvic, S. Srinivas, A.R. Tadipatri, T. Tao, V. Tsyrklevich, F. Vaquerizo-Villar, D. Weber, F. Zheng, ArXiv (n.d.).","mla":"Bolan, Matthew, et al. “The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale.” ArXiv, doi:10.48550/arXiv.2512.07087.","ieee":"M. Bolan et al., “The equational theories project: Advancing collaborative mathematical research at scale,” arXiv. .","apa":"Bolan, M., Breitner, J., Brox, J., Carlini, N., Carneiro, M., Doorn, F. van, … Zheng, F. (n.d.). The equational theories project: Advancing collaborative mathematical research at scale. arXiv. https://doi.org/10.48550/arXiv.2512.07087","ama":"Bolan M, Breitner J, Brox J, et al. The equational theories project: Advancing collaborative mathematical research at scale. arXiv. doi:10.48550/arXiv.2512.07087","ista":"Bolan M, Breitner J, Brox J, Carlini N, Carneiro M, Doorn F van, Dvorak M, Goens A, Hill A, Husum H, Mejia HI, Kocsis ZA, Floch BL, Bar-on A, Luccioli L, McNeil D, Meiburg A, Monticone P, Nielsen PP, Osazuwa EO, Paolini G, Petracci M, Reinke B, Renshaw D, Rossel M, Roux C, Scanvic J, Srinivas S, Tadipatri AR, Tao T, Tsyrklevich V, Vaquerizo-Villar F, Weber D, Zheng F. The equational theories project: Advancing collaborative mathematical research at scale. arXiv, 10.48550/arXiv.2512.07087.","chicago":"Bolan, Matthew, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, et al. “The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2512.07087."},"_id":"21399","month":"12","title":"The equational theories project: Advancing collaborative mathematical research at scale","date_created":"2026-03-04T12:00:16Z","department":[{"_id":"GradSch"},{"_id":"VlKo"}],"external_id":{"arxiv":["2512.07087"]},"OA_place":"repository","doi":"10.48550/arXiv.2512.07087","language":[{"iso":"eng"}],"arxiv":1,"publication_status":"submitted","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2512.07087"}],"type":"preprint","day":"16","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"oa":1,"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","author":[{"first_name":"Matthew","full_name":"Bolan, Matthew","last_name":"Bolan"},{"full_name":"Breitner, Joachim","last_name":"Breitner","first_name":"Joachim"},{"last_name":"Brox","full_name":"Brox, Jose","first_name":"Jose"},{"first_name":"Nicholas","full_name":"Carlini, Nicholas","last_name":"Carlini"},{"last_name":"Carneiro","full_name":"Carneiro, Mario","first_name":"Mario"},{"last_name":"Doorn","full_name":"Doorn, Floris van","first_name":"Floris van"},{"first_name":"Martin","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","full_name":"Dvorak, Martin","last_name":"Dvorak","orcid":"0000-0001-5293-214X"},{"last_name":"Goens","full_name":"Goens, Andrés","first_name":"Andrés"},{"full_name":"Hill, Aaron","last_name":"Hill","first_name":"Aaron"},{"full_name":"Husum, Harald","last_name":"Husum","first_name":"Harald"},{"last_name":"Mejia","full_name":"Mejia, Hernán Ibarra","first_name":"Hernán Ibarra"},{"last_name":"Kocsis","full_name":"Kocsis, Zoltan A.","first_name":"Zoltan A."},{"full_name":"Floch, Bruno Le","last_name":"Floch","first_name":"Bruno Le"},{"full_name":"Bar-on, Amir","last_name":"Bar-on","first_name":"Amir"},{"full_name":"Luccioli, Lorenzo","last_name":"Luccioli","first_name":"Lorenzo"},{"first_name":"Douglas","full_name":"McNeil, Douglas","last_name":"McNeil"},{"first_name":"Alex","last_name":"Meiburg","full_name":"Meiburg, Alex"},{"full_name":"Monticone, Pietro","last_name":"Monticone","first_name":"Pietro"},{"full_name":"Nielsen, Pace P.","last_name":"Nielsen","first_name":"Pace P."},{"full_name":"Osazuwa, Emmanuel Osalotioman","last_name":"Osazuwa","first_name":"Emmanuel Osalotioman"},{"full_name":"Paolini, Giovanni","last_name":"Paolini","first_name":"Giovanni"},{"full_name":"Petracci, Marco","last_name":"Petracci","first_name":"Marco"},{"last_name":"Reinke","full_name":"Reinke, Bernhard","first_name":"Bernhard"},{"last_name":"Renshaw","full_name":"Renshaw, David","first_name":"David"},{"full_name":"Rossel, Marcus","last_name":"Rossel","first_name":"Marcus"},{"first_name":"Cody","last_name":"Roux","full_name":"Roux, Cody"},{"first_name":"Jérémy","last_name":"Scanvic","full_name":"Scanvic, Jérémy"},{"first_name":"Shreyas","last_name":"Srinivas","full_name":"Srinivas, Shreyas"},{"first_name":"Anand Rao","last_name":"Tadipatri","full_name":"Tadipatri, Anand Rao"},{"first_name":"Terence","full_name":"Tao, Terence","last_name":"Tao"},{"last_name":"Tsyrklevich","full_name":"Tsyrklevich, Vlad","first_name":"Vlad"},{"last_name":"Vaquerizo-Villar","full_name":"Vaquerizo-Villar, Fernando","first_name":"Fernando"},{"first_name":"Daniel","full_name":"Weber, Daniel","last_name":"Weber"},{"first_name":"Fan","last_name":"Zheng","full_name":"Zheng, Fan"}],"publication":"arXiv","date_published":"2025-12-16T00:00:00Z","date_updated":"2026-03-12T08:36:21Z","year":"2025","oa_version":"Preprint"},{"date_created":"2026-03-04T11:56:29Z","external_id":{"arxiv":["2509.20539"]},"department":[{"_id":"GradSch"},{"_id":"VlKo"}],"doi":"10.48550/arXiv.2509.20539","corr_author":"1","OA_place":"repository","language":[{"iso":"eng"}],"arxiv":1,"publication_status":"draft","article_processing_charge":"No","abstract":[{"text":"Seymour's decomposition theorem is a hallmark result in matroid theory presenting a structural characterization of the class of regular matroids. Formalization of matroid theory faces many challenges, most importantly that only a limited number of notions and results have been implemented so far. In this work, we formalize the proof of the forward (composition) direction of Seymour's theorem for regular matroids. To this end, we develop a library in Lean 4 that implements definitions and results about totally unimodular matrices, vector matroids, their standard representations, regular matroids, and 1-, 2-, and 3-sums of matrices and binary matroids given by their standard representations. Using this framework, we formally state Seymour's decomposition theorem and implement a formally verified proof of the composition direction in the setting where the matroids have finite rank and may have infinite ground sets.","lang":"eng"}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"21393"}]},"citation":{"chicago":"Dvorak, Martin, Tristan Figueroa-Reid, Rida Hamadani, Byung-Hak Hwang, Evgenia Karunus, Vladimir Kolmogorov, Alexander Meiburg, et al. “Composition Direction of Seymour’s Theorem for Regular Matroids — Formally Verified.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2509.20539.","ista":"Dvorak M, Figueroa-Reid T, Hamadani R, Hwang B-H, Karunus E, Kolmogorov V, Meiburg A, Nelson A, Nelson P, Sandey M, Sergeev I. Composition direction of Seymour’s theorem for regular matroids — Formally verified. arXiv, 10.48550/arXiv.2509.20539.","ama":"Dvorak M, Figueroa-Reid T, Hamadani R, et al. Composition direction of Seymour’s theorem for regular matroids — Formally verified. arXiv. doi:10.48550/arXiv.2509.20539","apa":"Dvorak, M., Figueroa-Reid, T., Hamadani, R., Hwang, B.-H., Karunus, E., Kolmogorov, V., … Sergeev, I. (n.d.). Composition direction of Seymour’s theorem for regular matroids — Formally verified. arXiv. https://doi.org/10.48550/arXiv.2509.20539","ieee":"M. Dvorak et al., “Composition direction of Seymour’s theorem for regular matroids — Formally verified,” arXiv. .","short":"M. Dvorak, T. Figueroa-Reid, R. Hamadani, B.-H. Hwang, E. Karunus, V. Kolmogorov, A. Meiburg, A. Nelson, P. Nelson, M. Sandey, I. Sergeev, ArXiv (n.d.).","mla":"Dvorak, Martin, et al. “Composition Direction of Seymour’s Theorem for Regular Matroids — Formally Verified.” ArXiv, doi:10.48550/arXiv.2509.20539."},"_id":"21398","month":"09","title":"Composition direction of Seymour's theorem for regular matroids — Formally verified","publication":"arXiv","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","author":[{"orcid":"0000-0001-5293-214X","first_name":"Martin","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","full_name":"Dvorak, Martin","last_name":"Dvorak"},{"last_name":"Figueroa-Reid","full_name":"Figueroa-Reid, Tristan","first_name":"Tristan"},{"first_name":"Rida","last_name":"Hamadani","full_name":"Hamadani, Rida"},{"last_name":"Hwang","full_name":"Hwang, Byung-Hak","first_name":"Byung-Hak"},{"last_name":"Karunus","full_name":"Karunus, Evgenia","first_name":"Evgenia"},{"last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"},{"full_name":"Meiburg, Alexander","last_name":"Meiburg","first_name":"Alexander"},{"first_name":"Alexander","full_name":"Nelson, Alexander","last_name":"Nelson"},{"first_name":"Peter","last_name":"Nelson","full_name":"Nelson, Peter"},{"full_name":"Sandey, Mark","last_name":"Sandey","first_name":"Mark"},{"id":"ca3c9187-9a72-11ee-a009-8af825d896b0","first_name":"Ivan","full_name":"Sergeev, Ivan","last_name":"Sergeev","orcid":"0009-0004-9145-8785"}],"page":"21","date_published":"2025-09-23T00:00:00Z","date_updated":"2026-03-27T12:36:59Z","oa_version":"Preprint","year":"2025","day":"23","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2509.20539"}],"type":"preprint","acknowledgement":"We would like to dedicate the paper to the memory of Klaus Truemper, whose monograph Matroid Decomposition [12]\r\nlaid the foundation for our entire work.","status":"public","oa":1},{"_id":"17236","title":"A simpler and parallelizable O(√log n)-approximation algorithm for sparsest cut","month":"06","article_processing_charge":"Yes (via OA deal)","conference":{"name":"SPAA: Symposium on Parallelism in Algorithms and Architectures","start_date":"2024-06-17","location":"Nantes, France","end_date":"2024-06-21"},"citation":{"apa":"Kolmogorov, V. (2024). A simpler and parallelizable O(√log n)-approximation algorithm for sparsest cut. In Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures (pp. 403–414). Nantes, France: Association for Computing Machinery. https://doi.org/10.1145/3626183.3659969","ama":"Kolmogorov V. A simpler and parallelizable O(√log n)-approximation algorithm for sparsest cut. In: Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures. Association for Computing Machinery; 2024:403-414. doi:10.1145/3626183.3659969","ista":"Kolmogorov V. 2024. A simpler and parallelizable O(√log n)-approximation algorithm for sparsest cut. Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures. SPAA: Symposium on Parallelism in Algorithms and Architectures, 403–414.","chicago":"Kolmogorov, Vladimir. “A Simpler and Parallelizable O(√log n)-Approximation Algorithm for Sparsest Cut.” In Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures, 403–14. Association for Computing Machinery, 2024. https://doi.org/10.1145/3626183.3659969.","mla":"Kolmogorov, Vladimir. “A Simpler and Parallelizable O(√log n)-Approximation Algorithm for Sparsest Cut.” Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures, Association for Computing Machinery, 2024, pp. 403–14, doi:10.1145/3626183.3659969.","short":"V. Kolmogorov, in:, Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures, Association for Computing Machinery, 2024, pp. 403–414.","ieee":"V. Kolmogorov, “A simpler and parallelizable O(√log n)-approximation algorithm for sparsest cut,” in Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures, Nantes, France, 2024, pp. 403–414."},"related_material":{"record":[{"id":"21007","status":"public","relation":"extended_version"}]},"arxiv":1,"ddc":["510"],"doi":"10.1145/3626183.3659969","OA_place":"publisher","OA_type":"hybrid","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"day":"17","type":"conference","publisher":"Association for Computing Machinery","has_accepted_license":"1","date_published":"2024-06-17T00:00:00Z","year":"2024","oa_version":"Published Version","date_updated":"2026-01-21T09:46:25Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","page":"403-414","quality_controlled":"1","abstract":[{"lang":"eng","text":"Currently, the best known tradeoff between approximation ratio and complexity for the Sparsest Cut problem is achieved by the algorithm in [Sherman, FOCS 2009]: it computes O(√(log n)/ε)-approximation using O(nε logO(1) n) maxflows for any ε∈[Θ(1/log n),Θ(1)]. It works by solving the SDP relaxation of [Arora-Rao-Vazirani, STOC 2004] using the Multiplicative Weights Update algorithm (MW) of [Arora-Kale, JACM 2016]. To implement one MW step, Sherman approximately solves a multicommodity flow problem using another application of MW. Nested MW steps are solved via a certain \"chaining\" algorithm that combines results of multiple calls to the maxflow algorithm.\r\nWe present an alternative approach that avoids solving the multicommodity flow problem and instead computes \"violating paths\". This simplifies Sherman's algorithm by removing a need for a nested application of MW, and also allows parallelization: we show how to compute O(√(log n)/ε)-approximation via O(logO(1) n) maxflows using O(nε) processors.\r\nWe also revisit Sherman's chaining algorithm, and present a simpler version together with a new analysis."}],"publication_status":"published","external_id":{"isi":["001253331900044"],"arxiv":["2307.00115"]},"department":[{"_id":"VlKo"}],"date_created":"2024-07-14T22:01:11Z","language":[{"iso":"eng"}],"corr_author":"1","file":[{"success":1,"access_level":"open_access","relation":"main_file","file_size":1116166,"file_name":"2024_SPAA_Kolmogorov.pdf","creator":"dernst","date_created":"2024-07-16T06:38:08Z","file_id":"17245","date_updated":"2024-07-16T06:38:08Z","content_type":"application/pdf","checksum":"6ca18ac8508719dbd5d5735f4c991af2"}],"publication_identifier":{"issn":["1548-6109"],"isbn":["9798400704161"]},"oa":1,"file_date_updated":"2024-07-16T06:38:08Z","scopus_import":"1","publication":"Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures","author":[{"full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"}],"isi":1},{"arxiv":1,"OA_type":"green","doi":"10.48550/arXiv.2409.08119","OA_place":"repository","month":"09","title":"Duality theory in linear optimization and its extensions -- formally verified","_id":"20071","related_material":{"link":[{"relation":"software","url":"https://github.com/madvorak/duality/tree/v3.2","description":"full version of all definitions, statement, and proofs"}],"record":[{"relation":"dissertation_contains","status":"public","id":"21393"}]},"citation":{"ieee":"M. Dvorak and V. Kolmogorov, “Duality theory in linear optimization and its extensions -- formally verified,” arXiv. .","short":"M. Dvorak, V. Kolmogorov, ArXiv (n.d.).","mla":"Dvorak, Martin, and Vladimir Kolmogorov. “Duality Theory in Linear Optimization and Its Extensions -- Formally Verified.” ArXiv, 2409.08119, doi:10.48550/arXiv.2409.08119.","ista":"Dvorak M, Kolmogorov V. Duality theory in linear optimization and its extensions -- formally verified. arXiv, 2409.08119.","chicago":"Dvorak, Martin, and Vladimir Kolmogorov. “Duality Theory in Linear Optimization and Its Extensions -- Formally Verified.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2409.08119.","apa":"Dvorak, M., & Kolmogorov, V. (n.d.). Duality theory in linear optimization and its extensions -- formally verified. arXiv. https://doi.org/10.48550/arXiv.2409.08119","ama":"Dvorak M, Kolmogorov V. Duality theory in linear optimization and its extensions -- formally verified. arXiv. doi:10.48550/arXiv.2409.08119"},"article_processing_charge":"No","date_updated":"2026-03-27T12:36:59Z","oa_version":"Preprint","year":"2024","date_published":"2024-09-12T00:00:00Z","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","status":"public","type":"preprint","day":"12","publication_status":"draft","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2025-07-23T11:21:52Z","department":[{"_id":"GradSch"},{"_id":"VlKo"}],"external_id":{"arxiv":["2409.08119"]},"article_number":"2409.08119","abstract":[{"text":"Farkas established that a system of linear inequalities has a solution if and only if we cannot obtain a contradiction by taking a linear combination of the inequalities. We state and formally prove several Farkas-like theorems over linearly ordered fields in Lean 4. Furthermore, we extend duality theory to the case when some coefficients are allowed to take \"infinite values\".","lang":"eng"}],"publication":"arXiv","author":[{"id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","first_name":"Martin","last_name":"Dvorak","full_name":"Dvorak, Martin","orcid":"0000-0001-5293-214X"},{"last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"}],"keyword":["Farkas lemma","linear programming","extended reals","calculus of inductive constructions"],"oa":1,"acknowledgement":"We would like to thank David Bartl and Jasmin Blanchette for frequent consultations. We would also like to express gratitude to Andrew Yang for the proof of Finset.univ sum of zero when not and to Henrik B¨oving for a help with generalization from extended rationals to extended linearly ordered fields. We would also like to acknowledge Antoine Chambert-Loir, Apurva Nakade, Ya¨el Dillies, Richard Copley, Edward van de Meent, Markus Himmel, Mario Carneiro, and Kevin Buzzard.","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2409.08119"}]},{"file_date_updated":"2023-08-21T06:45:16Z","publication":"50th International Colloquium on Automata, Languages, and Programming","scopus_import":"1","author":[{"first_name":"David G.","last_name":"Harris","full_name":"Harris, David G."},{"full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir"}],"intvolume":" 261","acknowledgement":"We thank Heng Guo for helpful explanations of algorithms for sampling connected subgraphs and matchings, Maksym Serbyn for bringing to our attention the Wang-Landau algorithm and its use in physics.","file":[{"access_level":"open_access","relation":"main_file","success":1,"content_type":"application/pdf","checksum":"6dee0684245bb1c524b9c955db1e933d","date_updated":"2023-08-21T06:45:16Z","file_name":"2023_LIPIcsICALP_Harris.pdf","creator":"dernst","file_id":"14088","date_created":"2023-08-21T06:45:16Z","file_size":917791}],"publication_identifier":{"issn":["1868-8969"],"isbn":["9783959772785"]},"oa":1,"department":[{"_id":"VlKo"}],"external_id":{"arxiv":["2007.10824"]},"date_created":"2023-08-20T22:01:14Z","corr_author":"1","language":[{"iso":"eng"}],"publication_status":"published","abstract":[{"lang":"eng","text":"A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We will consider sampling problems which come from Gibbs distributions, which are families of probability distributions over a discrete space Ω with probability mass function of the form μ^Ω_β(ω) ∝ e^{β H(ω)} for β in an interval [β_min, β_max] and H(ω) ∈ {0} ∪ [1, n].\r\nThe partition function is the normalization factor Z(β) = ∑_{ω ∈ Ω} e^{β H(ω)}, and the log partition ratio is defined as q = (log Z(β_max))/Z(β_min)\r\nWe develop a number of algorithms to estimate the counts c_x using roughly Õ(q/ε²) samples for general Gibbs distributions and Õ(n²/ε²) samples for integer-valued distributions (ignoring some second-order terms and parameters), We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph."}],"article_number":"72","volume":261,"quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","date_published":"2023-07-01T00:00:00Z","oa_version":"Published Version","year":"2023","date_updated":"2025-07-10T11:50:45Z","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","type":"conference","day":"01","status":"public","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"doi":"10.4230/LIPIcs.ICALP.2023.72","alternative_title":["LIPIcs"],"arxiv":1,"ddc":["000","510"],"article_processing_charge":"Yes","conference":{"name":"ICALP: Automata, Languages and Programming","start_date":"2023-07-10","location":"Paderborn, Germany","end_date":"2023-07-14"},"citation":{"mla":"Harris, David G., and Vladimir Kolmogorov. “Parameter Estimation for Gibbs Distributions.” 50th International Colloquium on Automata, Languages, and Programming, vol. 261, 72, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023, doi:10.4230/LIPIcs.ICALP.2023.72.","short":"D.G. Harris, V. Kolmogorov, in:, 50th International Colloquium on Automata, Languages, and Programming, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023.","ieee":"D. G. Harris and V. Kolmogorov, “Parameter estimation for Gibbs distributions,” in 50th International Colloquium on Automata, Languages, and Programming, Paderborn, Germany, 2023, vol. 261.","ama":"Harris DG, Kolmogorov V. Parameter estimation for Gibbs distributions. In: 50th International Colloquium on Automata, Languages, and Programming. Vol 261. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2023. doi:10.4230/LIPIcs.ICALP.2023.72","apa":"Harris, D. G., & Kolmogorov, V. (2023). Parameter estimation for Gibbs distributions. In 50th International Colloquium on Automata, Languages, and Programming (Vol. 261). Paderborn, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ICALP.2023.72","chicago":"Harris, David G., and Vladimir Kolmogorov. “Parameter Estimation for Gibbs Distributions.” In 50th International Colloquium on Automata, Languages, and Programming, Vol. 261. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. https://doi.org/10.4230/LIPIcs.ICALP.2023.72.","ista":"Harris DG, Kolmogorov V. 2023. Parameter estimation for Gibbs distributions. 50th International Colloquium on Automata, Languages, and Programming. ICALP: Automata, Languages and Programming, LIPIcs, vol. 261, 72."},"related_material":{"record":[{"relation":"later_version","status":"public","id":"18855"}]},"_id":"14084","title":"Parameter estimation for Gibbs distributions","month":"07"},{"volume":2023,"quality_controlled":"1","abstract":[{"text":"We consider the problem of solving LP relaxations of MAP-MRF inference problems, and in particular the method proposed recently in [16], [35]. As a key computational subroutine, it uses a variant of the Frank-Wolfe (FW) method to minimize a smooth convex function over a combinatorial polytope. We propose an efficient implementation of this subroutine based on in-face Frank-Wolfe directions, introduced in [4] in a different context. More generally, we define an abstract data structure for a combinatorial subproblem that enables in-face FW directions, and describe its specialization for tree-structured MAP-MRF inference subproblems. Experimental results indicate that the resulting method is the current state-of-art LP solver for some classes of problems. Our code is available at pub.ist.ac.at/~vnk/papers/IN-FACE-FW.html.","lang":"eng"}],"publication_status":"published","external_id":{"isi":["001062522104029"],"arxiv":["2010.09567"]},"department":[{"_id":"VlKo"}],"date_created":"2023-10-22T22:01:16Z","language":[{"iso":"eng"}],"corr_author":"1","oa":1,"publication_identifier":{"issn":["1063-6919"],"isbn":["9798350301298"]},"main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2010.09567"}],"scopus_import":"1","author":[{"id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","first_name":"Vladimir","last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir"}],"publication":"Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition","intvolume":" 2023","isi":1,"_id":"14448","title":"Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions","month":"08","article_processing_charge":"No","conference":{"location":"Vancouver, Canada","end_date":"2023-06-24","name":"CVPR: Conference on Computer Vision and Pattern Recognition","start_date":"2023-06-17"},"citation":{"short":"V. Kolmogorov, in:, Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE, 2023, pp. 11980–11989.","mla":"Kolmogorov, Vladimir. “Solving Relaxations of MAP-MRF Problems: Combinatorial in-Face Frank-Wolfe Directions.” Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2023, IEEE, 2023, pp. 11980–89, doi:10.1109/CVPR52729.2023.01153.","ieee":"V. Kolmogorov, “Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vancouver, Canada, 2023, vol. 2023, pp. 11980–11989.","apa":"Kolmogorov, V. (2023). Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions. In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Vol. 2023, pp. 11980–11989). Vancouver, Canada: IEEE. https://doi.org/10.1109/CVPR52729.2023.01153","ama":"Kolmogorov V. Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions. In: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Vol 2023. IEEE; 2023:11980-11989. doi:10.1109/CVPR52729.2023.01153","ista":"Kolmogorov V. 2023. Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR: Conference on Computer Vision and Pattern Recognition vol. 2023, 11980–11989.","chicago":"Kolmogorov, Vladimir. “Solving Relaxations of MAP-MRF Problems: Combinatorial in-Face Frank-Wolfe Directions.” In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2023:11980–89. IEEE, 2023. https://doi.org/10.1109/CVPR52729.2023.01153."},"arxiv":1,"doi":"10.1109/CVPR52729.2023.01153","status":"public","publisher":"IEEE","day":"22","type":"conference","date_published":"2023-08-22T00:00:00Z","oa_version":"Preprint","year":"2023","date_updated":"2025-09-09T13:09:58Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","page":"11980-11989"},{"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","type":"conference","day":"27","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"status":"public","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-03-27T12:36:59Z","year":"2023","oa_version":"Published Version","has_accepted_license":"1","date_published":"2023-07-27T00:00:00Z","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"21393"}],"link":[{"url":"https://github.com/madvorak/grammars/tree/publish","relation":"software"}]},"citation":{"apa":"Dvorak, M., & Blanchette, J. (2023). Closure properties of general grammars - formally verified. In 14th International Conference on Interactive Theorem Proving (Vol. 268). Bialystok, Poland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ITP.2023.15","ama":"Dvorak M, Blanchette J. Closure properties of general grammars - formally verified. In: 14th International Conference on Interactive Theorem Proving. Vol 268. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2023. doi:10.4230/LIPIcs.ITP.2023.15","ista":"Dvorak M, Blanchette J. 2023. Closure properties of general grammars - formally verified. 14th International Conference on Interactive Theorem Proving. ITP: Interactive Theorem Proving, LIPIcs, vol. 268, 15.","chicago":"Dvorak, Martin, and Jasmin Blanchette. “Closure Properties of General Grammars - Formally Verified.” In 14th International Conference on Interactive Theorem Proving, Vol. 268. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. https://doi.org/10.4230/LIPIcs.ITP.2023.15.","short":"M. Dvorak, J. Blanchette, in:, 14th International Conference on Interactive Theorem Proving, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023.","mla":"Dvorak, Martin, and Jasmin Blanchette. “Closure Properties of General Grammars - Formally Verified.” 14th International Conference on Interactive Theorem Proving, vol. 268, 15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023, doi:10.4230/LIPIcs.ITP.2023.15.","ieee":"M. Dvorak and J. Blanchette, “Closure properties of general grammars - formally verified,” in 14th International Conference on Interactive Theorem Proving, Bialystok, Poland, 2023, vol. 268."},"conference":{"start_date":"2023-07-31","name":"ITP: Interactive Theorem Proving","end_date":"2023-08-04","location":"Bialystok, Poland"},"article_processing_charge":"No","month":"07","title":"Closure properties of general grammars - formally verified","_id":"13120","alternative_title":["LIPIcs"],"doi":"10.4230/LIPIcs.ITP.2023.15","ddc":["000"],"arxiv":1,"publication_identifier":{"isbn":["9783959772846"],"eissn":["1868-8969"]},"oa":1,"acknowledgement":"Jasmin Blanchette: This research has received funding from the Netherlands Organization\r\nfor Scientific Research (NWO) under the Vidi program (project No. 016.Vidi.189.037, Lean Forward).\r\n__\r\nWe thank Vladimir Kolmogorov for making this collaboration possible. We\r\nthank Václav Končický for discussing ideas about the Kleene star construction. We thank Patrick Johnson, Floris van Doorn, and Damiano Testa for their small yet very valuable contributions to our code. We thank Eric Wieser for simplifying one of our proofs. We thank Mark Summerfield for suggesting textual improvements. We thank the anonymous reviewers for very helpful comments. Finally, we thank the Lean community for helping us with various technical issues and answering many questions. ","file":[{"success":1,"access_level":"open_access","relation":"main_file","creator":"dernst","file_name":"2023_LIPIcS_Dvorak.pdf","file_id":"13982","date_created":"2023-08-07T11:55:43Z","file_size":715976,"checksum":"773a0197f05b67feaa6cb1e17ec3642d","content_type":"application/pdf","date_updated":"2023-08-07T11:55:43Z"}],"intvolume":" 268","isi":1,"scopus_import":"1","file_date_updated":"2023-08-07T11:55:43Z","author":[{"last_name":"Dvorak","full_name":"Dvorak, Martin","id":"40ED02A8-C8B4-11E9-A9C0-453BE6697425","first_name":"Martin","orcid":"0000-0001-5293-214X"},{"last_name":"Blanchette","full_name":"Blanchette, Jasmin","first_name":"Jasmin"}],"publication":"14th International Conference on Interactive Theorem Proving","article_number":"15","abstract":[{"text":"We formalized general (i.e., type-0) grammars using the Lean 3 proof assistant. We defined basic notions of rewrite rules and of words derived by a grammar, and used grammars to show closure of the class of type-0 languages under four operations: union, reversal, concatenation, and the Kleene star. The literature mostly focuses on Turing machine arguments, which are possibly more difficult to formalize. For the Kleene star, we could not follow the literature and came up with our own grammar-based construction.","lang":"eng"}],"quality_controlled":"1","volume":268,"corr_author":"1","language":[{"iso":"eng"}],"date_created":"2023-06-05T07:29:05Z","external_id":{"isi":["001515590500015"],"arxiv":["2302.06420"]},"department":[{"_id":"GradSch"},{"_id":"VlKo"}],"publication_status":"published"},{"publication_status":"published","date_created":"2022-02-06T23:01:32Z","external_id":{"isi":["000749997700015"],"arxiv":["1404.5475"]},"department":[{"_id":"VlKo"}],"language":[{"iso":"eng"}],"corr_author":"1","volume":26,"quality_controlled":"1","abstract":[{"text":"We consider two models for the sequence labeling (tagging) problem. The first one is a Pattern-Based Conditional Random Field (PB), in which the energy of a string (chain labeling) x=x1…xn∈Dn is a sum of terms over intervals [i,j] where each term is non-zero only if the substring xi…xj equals a prespecified word w∈Λ. The second model is a Weighted Context-Free Grammar (WCFG) frequently used for natural language processing. PB and WCFG encode local and non-local interactions respectively, and thus can be viewed as complementary. We propose a Grammatical Pattern-Based CRF model (GPB) that combines the two in a natural way. We argue that it has certain advantages over existing approaches such as the Hybrid model of Benedí and Sanchez that combines N-grams and WCFGs. The focus of this paper is to analyze the complexity of inference tasks in a GPB such as computing MAP. We present a polynomial-time algorithm for general GPBs and a faster version for a special case that we call Interaction Grammars.","lang":"eng"}],"publication":"Intelligent Data Analysis","scopus_import":"1","author":[{"id":"2CCAC26C-F248-11E8-B48F-1D18A9856A87","first_name":"Rustem","last_name":"Takhanov","full_name":"Takhanov, Rustem"},{"full_name":"Kolmogorov, Vladimir","last_name":"Kolmogorov","first_name":"Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87"}],"isi":1,"intvolume":" 26","publication_identifier":{"eissn":["1571-4128"],"issn":["1088-467X"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1404.5475"}],"arxiv":1,"article_type":"original","doi":"10.3233/IDA-205623","_id":"10737","month":"01","title":"Combining pattern-based CRFs and weighted context-free grammars","article_processing_charge":"No","citation":{"ista":"Takhanov R, Kolmogorov V. 2022. Combining pattern-based CRFs and weighted context-free grammars. Intelligent Data Analysis. 26(1), 257–272.","chicago":"Takhanov, Rustem, and Vladimir Kolmogorov. “Combining Pattern-Based CRFs and Weighted Context-Free Grammars.” Intelligent Data Analysis. IOS Press, 2022. https://doi.org/10.3233/IDA-205623.","apa":"Takhanov, R., & Kolmogorov, V. (2022). Combining pattern-based CRFs and weighted context-free grammars. Intelligent Data Analysis. IOS Press. https://doi.org/10.3233/IDA-205623","ama":"Takhanov R, Kolmogorov V. Combining pattern-based CRFs and weighted context-free grammars. Intelligent Data Analysis. 2022;26(1):257-272. doi:10.3233/IDA-205623","ieee":"R. Takhanov and V. Kolmogorov, “Combining pattern-based CRFs and weighted context-free grammars,” Intelligent Data Analysis, vol. 26, no. 1. IOS Press, pp. 257–272, 2022.","short":"R. Takhanov, V. Kolmogorov, Intelligent Data Analysis 26 (2022) 257–272.","mla":"Takhanov, Rustem, and Vladimir Kolmogorov. “Combining Pattern-Based CRFs and Weighted Context-Free Grammars.” Intelligent Data Analysis, vol. 26, no. 1, IOS Press, 2022, pp. 257–72, doi:10.3233/IDA-205623."},"date_published":"2022-01-14T00:00:00Z","date_updated":"2024-10-09T21:01:33Z","year":"2022","oa_version":"Preprint","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","page":"257-272","status":"public","issue":"1","type":"journal_article","day":"14","publisher":"IOS Press"},{"language":[{"iso":"eng"}],"date_created":"2021-05-02T22:01:29Z","external_id":{"isi":["000640109300001"]},"department":[{"_id":"VlKo"}],"publication_status":"published","abstract":[{"text":"In this paper, we propose a new iterative method with alternated inertial step for solving split common null point problem in real Hilbert spaces. We obtain weak convergence of the proposed iterative algorithm. Furthermore, we introduce the notion of bounded linear regularity property for the split common null point problem and obtain the linear convergence property for the new algorithm under some mild assumptions. Finally, we provide some numerical examples to demonstrate the performance and efficiency of the proposed method.","lang":"eng"}],"quality_controlled":"1","volume":71,"isi":1,"intvolume":" 71","ec_funded":1,"author":[{"first_name":"Ferdinard U.","last_name":"Ogbuisi","full_name":"Ogbuisi, Ferdinard U."},{"orcid":"0000-0001-9224-7139","first_name":"Yekini","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","last_name":"Shehu","full_name":"Shehu, Yekini"},{"last_name":"Yao","full_name":"Yao, Jen Chih","first_name":"Jen Chih"}],"scopus_import":"1","publication":"Optimization","publication_identifier":{"issn":["0233-1934"],"eissn":["1029-4945"]},"acknowledgement":"The second author has received funding from the European Research Council (ERC) under the European Union's Seventh Framework Program (FP7-2007-2013) (Grant agreement No. 616160).","article_type":"original","doi":"10.1080/02331934.2021.1914035","citation":{"chicago":"Ogbuisi, Ferdinard U., Yekini Shehu, and Jen Chih Yao. “Convergence Analysis of New Inertial Method for the Split Common Null Point Problem.” Optimization. Taylor and Francis, 2022. https://doi.org/10.1080/02331934.2021.1914035.","ista":"Ogbuisi FU, Shehu Y, Yao JC. 2022. Convergence analysis of new inertial method for the split common null point problem. Optimization. 71(13), 3767–3795.","ama":"Ogbuisi FU, Shehu Y, Yao JC. Convergence analysis of new inertial method for the split common null point problem. Optimization. 2022;71(13):3767-3795. doi:10.1080/02331934.2021.1914035","apa":"Ogbuisi, F. U., Shehu, Y., & Yao, J. C. (2022). Convergence analysis of new inertial method for the split common null point problem. Optimization. Taylor and Francis. https://doi.org/10.1080/02331934.2021.1914035","ieee":"F. U. Ogbuisi, Y. Shehu, and J. C. Yao, “Convergence analysis of new inertial method for the split common null point problem,” Optimization, vol. 71, no. 13. Taylor and Francis, pp. 3767–3795, 2022.","mla":"Ogbuisi, Ferdinard U., et al. “Convergence Analysis of New Inertial Method for the Split Common Null Point Problem.” Optimization, vol. 71, no. 13, Taylor and Francis, 2022, pp. 3767–95, doi:10.1080/02331934.2021.1914035.","short":"F.U. Ogbuisi, Y. Shehu, J.C. Yao, Optimization 71 (2022) 3767–3795."},"article_processing_charge":"No","month":"11","title":"Convergence analysis of new inertial method for the split common null point problem","_id":"9365","page":"3767-3795","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_updated":"2024-11-04T13:52:36Z","oa_version":"None","year":"2022","project":[{"grant_number":"616160","_id":"25FBA906-B435-11E9-9278-68D0E5697425","name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7"}],"date_published":"2022-11-01T00:00:00Z","issue":"13","publisher":"Taylor and Francis","day":"01","type":"journal_article","status":"public"},{"abstract":[{"lang":"eng","text":"In this paper, we consider reflected three-operator splitting methods for monotone inclusion problems in real Hilbert spaces. To do this, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the reflected Krasnosel'skiĭ-Mann iteration for finding a fixed point of nonexpansive mapping in real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. We then apply our results to three-operator splitting for the monotone inclusion problem and consequently obtain the corresponding convergence analysis. Furthermore, we derive reflected primal-dual algorithms for highly structured monotone inclusion problems. Some numerical implementations are drawn from splitting methods to support the theoretical analysis."}],"quality_controlled":"1","volume":37,"language":[{"iso":"eng"}],"corr_author":"1","date_created":"2021-06-06T22:01:30Z","external_id":{"isi":["000650507600001"]},"department":[{"_id":"VlKo"}],"publication_status":"published","publication_identifier":{"issn":["1055-6788"],"eissn":["1029-4937"]},"acknowledgement":"The authors are grateful to the anonymous referees and the handling Editor for their insightful comments which have improved the earlier version of the manuscript greatly. The second author is grateful to the University of Hafr Al Batin. The last author has received funding from the European Research Council (ERC) under the European Union's Seventh Framework Program (FP7-2007-2013) (Grant agreement No. 616160).","isi":1,"intvolume":" 37","ec_funded":1,"publication":"Optimization Methods and Software","author":[{"full_name":"Iyiola, Olaniyi S.","last_name":"Iyiola","first_name":"Olaniyi S."},{"last_name":"Enyi","full_name":"Enyi, Cyril D.","first_name":"Cyril D."},{"orcid":"0000-0001-9224-7139","first_name":"Yekini","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87","last_name":"Shehu","full_name":"Shehu, Yekini"}],"scopus_import":"1","citation":{"apa":"Iyiola, O. S., Enyi, C. D., & Shehu, Y. (2022). Reflected three-operator splitting method for monotone inclusion problem. Optimization Methods and Software. Taylor and Francis. https://doi.org/10.1080/10556788.2021.1924715","ama":"Iyiola OS, Enyi CD, Shehu Y. Reflected three-operator splitting method for monotone inclusion problem. Optimization Methods and Software. 2022;37(4):1527-1565. doi:10.1080/10556788.2021.1924715","ista":"Iyiola OS, Enyi CD, Shehu Y. 2022. Reflected three-operator splitting method for monotone inclusion problem. Optimization Methods and Software. 37(4), 1527–1565.","chicago":"Iyiola, Olaniyi S., Cyril D. Enyi, and Yekini Shehu. “Reflected Three-Operator Splitting Method for Monotone Inclusion Problem.” Optimization Methods and Software. Taylor and Francis, 2022. https://doi.org/10.1080/10556788.2021.1924715.","mla":"Iyiola, Olaniyi S., et al. “Reflected Three-Operator Splitting Method for Monotone Inclusion Problem.” Optimization Methods and Software, vol. 37, no. 4, Taylor and Francis, 2022, pp. 1527–65, doi:10.1080/10556788.2021.1924715.","short":"O.S. Iyiola, C.D. Enyi, Y. Shehu, Optimization Methods and Software 37 (2022) 1527–1565.","ieee":"O. S. Iyiola, C. D. Enyi, and Y. Shehu, “Reflected three-operator splitting method for monotone inclusion problem,” Optimization Methods and Software, vol. 37, no. 4. Taylor and Francis, pp. 1527–1565, 2022."},"article_processing_charge":"No","month":"07","title":"Reflected three-operator splitting method for monotone inclusion problem","_id":"9469","article_type":"original","doi":"10.1080/10556788.2021.1924715","issue":"4","day":"01","type":"journal_article","publisher":"Taylor and Francis","status":"public","page":"1527-1565","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","date_updated":"2024-11-04T13:52:36Z","year":"2022","oa_version":"None","date_published":"2022-07-01T00:00:00Z","project":[{"call_identifier":"FP7","name":"Discrete Optimization in Computer Vision: Theory and Practice","_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160"}]},{"ddc":["510","515","518"],"arxiv":1,"article_type":"original","doi":"10.1080/00036811.2020.1736287","month":"01","title":"Weak convergence for variational inequalities with inertial-type method","_id":"7577","citation":{"short":"Y. Shehu, O.S. Iyiola, Applicable Analysis 101 (2022) 192–216.","mla":"Shehu, Yekini, and Olaniyi S. Iyiola. “Weak Convergence for Variational Inequalities with Inertial-Type Method.” Applicable Analysis, vol. 101, no. 1, Taylor & Francis, 2022, pp. 192–216, doi:10.1080/00036811.2020.1736287.","ieee":"Y. Shehu and O. S. Iyiola, “Weak convergence for variational inequalities with inertial-type method,” Applicable Analysis, vol. 101, no. 1. Taylor & Francis, pp. 192–216, 2022.","apa":"Shehu, Y., & Iyiola, O. S. (2022). Weak convergence for variational inequalities with inertial-type method. Applicable Analysis. Taylor & Francis. https://doi.org/10.1080/00036811.2020.1736287","ama":"Shehu Y, Iyiola OS. Weak convergence for variational inequalities with inertial-type method. Applicable Analysis. 2022;101(1):192-216. doi:10.1080/00036811.2020.1736287","ista":"Shehu Y, Iyiola OS. 2022. Weak convergence for variational inequalities with inertial-type method. Applicable Analysis. 101(1), 192–216.","chicago":"Shehu, Yekini, and Olaniyi S. Iyiola. “Weak Convergence for Variational Inequalities with Inertial-Type Method.” Applicable Analysis. Taylor & Francis, 2022. https://doi.org/10.1080/00036811.2020.1736287."},"article_processing_charge":"No","date_updated":"2024-11-04T13:52:44Z","year":"2022","oa_version":"Submitted Version","project":[{"_id":"25FBA906-B435-11E9-9278-68D0E5697425","grant_number":"616160","name":"Discrete Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7"}],"date_published":"2022-01-01T00:00:00Z","has_accepted_license":"1","page":"192-216","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","issue":"1","publisher":"Taylor & Francis","day":"01","type":"journal_article","publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2020-03-09T07:06:52Z","department":[{"_id":"VlKo"}],"external_id":{"isi":["000518364100001"],"arxiv":["2101.08057"]},"quality_controlled":"1","volume":101,"abstract":[{"lang":"eng","text":"Weak convergence of inertial iterative method for solving variational inequalities is the focus of this paper. The cost function is assumed to be non-Lipschitz and monotone. We propose a projection-type method with inertial terms and give weak convergence analysis under appropriate conditions. Some test results are performed and compared with relevant methods in the literature to show the efficiency and advantages given by our proposed methods."}],"isi":1,"intvolume":" 101","ec_funded":1,"scopus_import":"1","publication":"Applicable Analysis","file_date_updated":"2021-03-16T23:30:06Z","author":[{"orcid":"0000-0001-9224-7139","last_name":"Shehu","full_name":"Shehu, Yekini","first_name":"Yekini","id":"3FC7CB58-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Iyiola","full_name":"Iyiola, Olaniyi S.","first_name":"Olaniyi S."}],"oa":1,"publication_identifier":{"eissn":["1563-504X"],"issn":["0003-6811"]},"file":[{"file_size":4282586,"file_name":"2020_ApplicAnalysis_Shehu.pdf","creator":"dernst","date_created":"2020-10-12T10:42:54Z","file_id":"8648","date_updated":"2021-03-16T23:30:06Z","content_type":"application/pdf","checksum":"869efe8cb09505dfa6012f67d20db63d","embargo":"2021-03-15","access_level":"open_access","relation":"main_file"}],"acknowledgement":"The project of the first author has received funding from the European Research Council (ERC) under the European Union's Seventh Framework Program (FP7 - 2007-2013) (Grant agreement No. 616160)."},{"arxiv":1,"month":"07","title":"One-sided Frank-Wolfe algorithms for saddle problems","_id":"10552","citation":{"ama":"Kolmogorov V, Pock T. One-sided Frank-Wolfe algorithms for saddle problems. In: 38th International Conference on Machine Learning. ; 2021.","apa":"Kolmogorov, V., & Pock, T. (2021). One-sided Frank-Wolfe algorithms for saddle problems. In 38th International Conference on Machine Learning. Virtual.","chicago":"Kolmogorov, Vladimir, and Thomas Pock. “One-Sided Frank-Wolfe Algorithms for Saddle Problems.” In 38th International Conference on Machine Learning, 2021.","ista":"Kolmogorov V, Pock T. 2021. One-sided Frank-Wolfe algorithms for saddle problems. 38th International Conference on Machine Learning. ICML: International Conference on Machine Learning.","mla":"Kolmogorov, Vladimir, and Thomas Pock. “One-Sided Frank-Wolfe Algorithms for Saddle Problems.” 38th International Conference on Machine Learning, 2021.","short":"V. Kolmogorov, T. Pock, in:, 38th International Conference on Machine Learning, 2021.","ieee":"V. Kolmogorov and T. Pock, “One-sided Frank-Wolfe algorithms for saddle problems,” in 38th International Conference on Machine Learning, Virtual, 2021."},"conference":{"start_date":"2021-07-18","name":"ICML: International Conference on Machine Learning","end_date":"2021-07-24","location":"Virtual"},"article_processing_charge":"No","date_updated":"2025-07-16T08:04:23Z","oa_version":"Preprint","year":"2021","date_published":"2021-07-01T00:00:00Z","project":[{"call_identifier":"FP7","name":"Discrete Optimization in Computer Vision: Theory and Practice","grant_number":"616160","_id":"25FBA906-B435-11E9-9278-68D0E5697425"}],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","status":"public","day":"01","type":"conference","publication_status":"published","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2021-12-16T12:41:20Z","external_id":{"arxiv":["2101.12617"]},"department":[{"_id":"VlKo"}],"quality_controlled":"1","abstract":[{"lang":"eng","text":"We study a class of convex-concave saddle-point problems of the form minxmaxy⟨Kx,y⟩+fP(x)−h∗(y) where K is a linear operator, fP is the sum of a convex function f with a Lipschitz-continuous gradient and the indicator function of a bounded convex polytope P, and h∗ is a convex (possibly nonsmooth) function. Such problem arises, for example, as a Lagrangian relaxation of various discrete optimization problems. Our main assumptions are the existence of an efficient linear minimization oracle (lmo) for fP and an efficient proximal map for h∗ which motivate the solution via a blend of proximal primal-dual algorithms and Frank-Wolfe algorithms. In case h∗ is the indicator function of a linear constraint and function f is quadratic, we show a O(1/n2) convergence rate on the dual objective, requiring O(nlogn) calls of lmo. If the problem comes from the constrained optimization problem minx∈Rd{fP(x)|Ax−b=0} then we additionally get bound O(1/n2) both on the primal gap and on the infeasibility gap. In the most general case, we show a O(1/n) convergence rate of the primal-dual gap again requiring O(nlogn) calls of lmo. To the best of our knowledge, this improves on the known convergence rates for the considered class of saddle-point problems. We show applications to labeling problems frequently appearing in machine learning and computer vision."}],"ec_funded":1,"publication":"38th International Conference on Machine Learning","author":[{"first_name":"Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","last_name":"Kolmogorov","full_name":"Kolmogorov, Vladimir"},{"first_name":"Thomas","full_name":"Pock, Thomas","last_name":"Pock"}],"scopus_import":"1","oa":1,"acknowledgement":"Vladimir Kolmogorov was supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160. Thomas Pock acknowledges support by an ERC grant HOMOVIS, no 640156.","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2101.12617"}]}]