@inproceedings{22617,
  abstract     = {Consider a 4-player version of Matching Pennies where a team of three players competes against the Devil. Each player simultaneously says "Heads" or "Tails". The team wins if all four choices match; otherwise the Devil wins. If all team players randomise independently, they win with probability 1/8; if all players share a common source of randomness, they win with probability 1/2. What happens when each pair of team players shares a source of randomness? Can the team do better than win with probability 1/4? The surprising (and nontrivial) answer is yes!
We introduce Dicey Games, a formal framework motivated by the study of distributed systems with shared sources of randomness (of which the above example is a specific instance). We characterise the existence, representation and computational complexity of optimal strategies in Dicey Games, and we study the problem of allocating limited sources of randomness optimally within a team.},
  author       = {Brice, Leonard J and Henzinger, Thomas A and Thejaswini, K. S.},
  booktitle    = {41st Annual Symposium on Logic in Computer Science},
  isbn         = {9783959774345},
  issn         = {1868-8969},
  keywords     = {Concurrent games, Shared randomness, Topology, Algebraic Geometry},
  location     = {Lisbon, Portugal},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Dicey games: Shared sources of randomness in distributed systems}},
  doi          = {10.4230/LIPIcs.LICS.2026.23},
  volume       = {380},
  year         = {2026},
}

@inproceedings{22916,
  abstract     = {In the imprecise geometry model, the input is a family of regions F = (R₁, R₂, …,R_n), each containing a point p_i ∈ R_i. The task is then to compute some function of the points p₁,p₂,… p_n, in our case an implicit representation of their Pareto front. To this end, one may query a region R_i to retrieve its contained point p_i ∈ R_i. In this model, efficiency is interpreted in two ways: minimizing (i) the number of retrievals, and (ii) the computation time both for preprocessing, and the execution of the query stage, i.e. for computing which points to query and constructing the output. 
We present an algorithm to construct (an implicit representation of) the Pareto front for possibly overlapping rectangles, that is instance-optimal with respect to the number of retrievals. This means that for every fixed input (F, P), there is no algorithm that retrieves asymptotically fewer regions to compute the output. This is a strong algorithmic quality, as it means that our algorithm is competitive even to clairvoyant algorithms which only have to verify the correctness of a correct guess. In terms of algorithmic running time, instance-optimality is provably unobtainable. We instead present an algorithm which is within a log n-factor of instance optimality. This generalizes earlier results which assumed the regions to not overlap, at only a minor cost in running time. 
For unit squares, we present an algorithm that is not only instance optimal in the number of retrievals, but also universally optimal in terms of running time. This means that for any fixed set of regions F, no algorithm has a better worst-case running time for all possible point sets P. Thus, this work presents the first universally optimal algorithm for overlapping planar input. Compared to previous work, our result improves the degree to which the input regions may overlap, the preprocessing time, the number of retrievals, and the running time.},
  author       = {De Berg, Sarita and Bække, Nynne Maria Foldager and Eriksen, Frida Astrup and Van Der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel P},
  booktitle    = {34th Annual European Symposium on Algorithms},
  isbn         = {9783959774451},
  issn         = {1868-8969},
  keywords     = {Pareto front, imprecise geometry, instance optimality, universal optimality, preprocessing model, partial information},
  location     = {L’Aquila, Italy},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Instance optimal and universally optimal bounds for imprecise pareto fronts}},
  doi          = {10.4230/LIPIcs.ESA.2026.106},
  volume       = {388},
  year         = {2026},
}

@inproceedings{22915,
  abstract     = {The element distinctness problem takes as input a list I of n values from a totally ordered universe, where pairwise comparisons between values are allowed, and the goal is to decide whether I contains any duplicates. It is a well-studied problem with a classical worst-case Ω(n log n) comparison-based lower bound by Fredman [TCS'76]. At first glance, this lower bound appears to rule out any algorithm more efficient than the naive approach of sorting I and comparing adjacent elements. However, upon closer inspection, the Ω(n log n) bound is overly pessimistic. For instance, if I contains n/2 identical elements, a median-finding algorithm will, regardless of the input order, find a duplicate in linear time. This raises a natural question: Are there comparison-based lower bounds for element distinctness that are sensitive to the amount of duplicates in the input instance?
To address this question, we derive instance-specific lower bounds. For any input instance I, we represent the combinatorial structure of the duplicates in I by an undirected graph G(I) that connects identical elements. Each such graph G is a union of cliques, and we study algorithms by their worst-case running time over all inputs I' with G(I') ≅ G. We establish an adversarial lower bound showing that, for any deterministic algorithm 𝒜, there exists a graph G and an algorithm 𝒜' that, for all inputs I with G(I) ≅ G, is a factor O(log log n) faster than 𝒜. Consequently, no deterministic algorithm can be o(log log n)-competitive for all graphs G. We complement this with an O(log log n)-competitive deterministic algorithm, thereby obtaining tight bounds for element distinctness that go beyond classical worst-case analysis. Subsequently, we study the related problem of set intersection. We show that no deterministic set intersection algorithm can be o(log n)-competitive, and provide an O(log n)-competitive deterministic algorithm. We find it interesting and surprising to discover tight O(log log n)-competitive bounds for element distinctness. Moreover, we find the separation between element distinctness and the set intersection problem unexpected.},
  author       = {Van Der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel P},
  booktitle    = {34th Annual European Symposium on Algorithms},
  isbn         = {9783959774451},
  issn         = {1868-8969},
  keywords     = {Comparison-based analysis, set intersection, universal optimality},
  location     = {L’Aquila, Italy},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Tight Better-Than-Worst-Case bounds for element distinctness and set intersection}},
  doi          = {10.4230/LIPIcs.ESA.2026.101},
  volume       = {388},
  year         = {2026},
}

@inproceedings{22918,
  abstract     = {Given a weighted digraph G, a (t,g,μ)-DAG cover is a collection of g dominating DAGs D_1,… ,D_g such that all distances are approximately preserved: for every pair (u,v) of vertices, min_id_{D_i}(u,v) ≤ t⋅ d_G(u,v), and the total number of non-G edges is bounded by |(∪_i D_i)⧵ G| ≤ μ. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of Steiner DAG cover, where the DAGs are allowed to contain Steiner points. 
We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a (1,2,Õ(n⋅tw))-Steiner DAG cover. For planar digraphs we provide a (1+ε,2,Õ_ε(n))-Steiner DAG cover.
We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth 1 with stretch t < 2 and sub-quadratic number of extra edges requires Ω(log n) DAGs.},
  author       = {Bhore, Sujoy and Chang, Hsien Chih and Conroy, Jonathan and Filtser, Arnold and Oh, Eunjin and Wein, Nicole and Zheng, Da Wei},
  booktitle    = {34th Annual European Symposium on Algorithms},
  isbn         = {9783959774451},
  issn         = {1868-8969},
  keywords     = {Directed graphs, DAG (directed acyclic graphs), distortion, metric embeddings, planar graph, treewidth},
  location     = {L’Aquila, Italy},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{DAG covers for structured graphs: The Steiner point effect}},
  doi          = {10.4230/LIPIcs.ESA.2026.94},
  volume       = {388},
  year         = {2026},
}

@inproceedings{22921,
  abstract     = {Runtime monitoring of quantitative signals faces a fundamental trade-off between volatility and over-aggregation: instantaneous observations are noisy, while long-run averages obscure local structure. Localisation measures such as discounted averages offer a principled middle ground, yet remain poorly understood in runtime verification. This paper studies discounted sums from a monitoring perspective, in both deterministic and stochastic settings. We formalize the discounted monitoring problem and show that exact, sound monitoring of discounted sums cannot be achieved with finite memory. To overcome this impossibility, we introduce ε-approximately sound monitoring, deriving explicit bounds on memory and observation requirements. We then extend the framework to stochastic processes via expected discounted sums, defining pointwise and uniform (ε,δ)-soundness notions, establishing statistical optimality, and proving impossibility beyond a precision threshold. We also formalize the resource complexity of deterministic discounted monitoring via affine register machines and prove a tight worst-case lower bound. Finally, we present a specification language for arithmetic expressions over multiple discounted sums with synchronous and asynchronous semantics, and evaluate our approach on practical scenarios including algorithmic fairness.},
  author       = {Cano Cordoba, Filip and Henzinger, Thomas A and Kueffner, Konstantin and Sarac, Naci E},
  booktitle    = {37th International Conference on Concurrency Theory},
  isbn         = {9783959774475},
  issn         = {1868-8969},
  keywords     = {Runtime Verification, Probabilistic Systems, Quantitative Verification, Approximate Monitoring},
  location     = {Liverpool, United Kingdom},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Monitoring discounted sum properties}},
  doi          = {10.4230/LIPIcs.CONCUR.2026.22},
  volume       = {391},
  year         = {2026},
}

@inproceedings{22922,
  abstract     = {In this note, we recall the history and our motivation behind the development of Strategy Logic and we discuss some of the work that ensued from its introduction.},
  author       = {Chatterjee, Krishnendu and Henzinger, Thomas A and Piterman, Nir},
  booktitle    = {37th International Conference on Concurrency Theory},
  isbn         = {9783959774475},
  issn         = {1868-8969},
  keywords     = {Strategy Logic, Games, Automata},
  location     = {Liverpool, United Kingdom},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{A look back at strategy logic}},
  doi          = {10.4230/LIPIcs.CONCUR.2026.6},
  volume       = {391},
  year         = {2026},
}

@inproceedings{19281,
  abstract     = {In this work, we consider the list-decodability and list-recoverability of codes in the zero-rate regime. Briefly, a code 𝒞 ⊆ [q]ⁿ is (p,𝓁,L)-list-recoverable if for all tuples of input lists (Y₁,… ,Y_n) with each Y_i ⊆ [q] and |Y_i| = 𝓁, the number of codewords c ∈ 𝒞 such that c_i ∉ Y_i for at most pn choices of i ∈ [n] is less than L; list-decoding is the special case of 𝓁 = 1. In recent work by Resch, Yuan and Zhang (ICALP 2023) the zero-rate threshold for list-recovery was determined for all parameters: that is, the work explicitly computes p_*: = p_*(q,𝓁,L) with the property that for all ε > 0 (a) there exist positive-rate (p_*-ε,𝓁,L)-list-recoverable codes, and (b) any (p_*+ε,𝓁,L)-list-recoverable code has rate 0. In fact, in the latter case the code has constant size, independent on n. However, the constant size in their work is quite large in 1/ε, at least |𝒞| ≥ (1/(ε))^O(q^L).
Our contribution in this work is to show that for all choices of q,𝓁 and L with q ≥ 3, any (p_*+ε,𝓁,L)-list-recoverable code must have size O_{q,𝓁,L}(1/ε), and furthermore this upper bound is complemented by a matching lower bound Ω_{q,𝓁,L}(1/ε). This greatly generalizes work by Alon, Bukh and Polyanskiy (IEEE Trans. Inf. Theory 2018) which focused only on the case of binary alphabet (and thus necessarily only list-decoding). We remark that we can in fact recover the same result for q = 2 and even L, as obtained by Alon, Bukh and Polyanskiy: we thus strictly generalize their work. 
Our main technical contribution is to (a) properly define a linear programming relaxation of the list-recovery condition over large alphabets; and (b) to demonstrate that a certain function defined on a q-ary probability simplex is maximized by the uniform distribution. This represents the core challenge in generalizing to larger q (as a binary simplex can be naturally identified with a one-dimensional interval). We can subsequently re-utilize certain Schur convexity and convexity properties established for a related function by Resch, Yuan and Zhang along with ideas of Alon, Bukh and Polyanskiy.},
  author       = {Resch, Nicolas and Yuan, Chen and Zhang, Yihan},
  booktitle    = {16th Innovations in Theoretical Computer Science Conference},
  isbn         = {9783959773614},
  issn         = {1868-8969},
  location     = {New York, NY, United States},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Tight bounds on list-decodable and list-recoverable zero-rate codes}},
  doi          = {10.4230/LIPIcs.ITCS.2025.82},
  volume       = {325},
  year         = {2025},
}

@inproceedings{19858,
  abstract     = {Given a graph G that undergoes a sequence of edge insertions and deletions, we study the Maximum k-Edge Coloring problem (MkEC): Having access to k different colors, color as many edges of G as possible such that no two adjacent edges share the same color. While this problem is different from simply maintaining a b-matching with b = k, the two problems are related. However, maximum b-matching can be solved efficiently in the static setting, whereas MkEC is NP-hard and even APX-hard for k ≥ 2. 
We present new results on both problems: For b-matching, we show a new integrality gap result and we adapt Wajc’s matching sparsification scheme [David Wajc, 2020] for the case where b is a constant.
Using these as basis, we give three new algorithms for the dynamic MkEC problem: Our MatchO algorithm builds on the dynamic (2+ε)-approximation algorithm of Bhattacharya, Gupta, and Mohan [Sayan Bhattacharya et al., 2017] for b-matching and achieves a (2+ε)(k+1)/k-approximation in O(poly(log n, ε^-1)) update time against an oblivious adversary. Our MatchA algorithm builds on the dynamic (7+ε)-approximation algorithm by Bhattacharya, Henzinger, and Italiano [Sayan Bhattacharya et al., 2015] for fractional b-matching and achieves a (7+ε)(3k+3)/(3k-1)-approximation in O(poly(log n, ε^-1)) update time against an adaptive adversary. Moreover, our reductions use the dynamic b-matching algorithm as a black box, so any future improvement in the approximation ratio for dynamic b-matching will automatically translate into a better approximation ratio for our algorithms. Finally, we present a greedy algorithm with O(Δ+k) update time, which guarantees a 2.16 approximation factor.},
  author       = {El-Hayek, Antoine and Hanauer, Kathrin and Henzinger, Monika H},
  booktitle    = {4th Symposium on Algorithmic Foundations of Dynamic Networks},
  isbn         = {9783959773683},
  issn         = {1868-8969},
  location     = {Liverpool, United Kingdom},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{On b-matching and fully-dynamic maximum k-edge coloring}},
  doi          = {10.4230/LIPIcs.SAND.2025.4},
  volume       = {330},
  year         = {2025},
}

@inproceedings{20253,
  abstract     = {A quantitative word automaton (QWA) defines a function from infinite words to values. For example, every infinite run of a limit-average QWA 𝒜 obtains a mean payoff, and every word w ∈ Σ^ω is assigned the maximal mean payoff obtained by nondeterministic runs of 𝒜 over w. We introduce quantitative language automata (QLAs) that define functions from language generators (i.e., implementations) to values, where a language generator can be nonprobabilistic, defining a set of infinite words, or probabilistic, defining a probability measure over infinite words. A QLA consists of a QWA and an aggregator function. For example, given a QWA 𝒜, the infimum aggregator maps each language L ⊆ Σ^ω to the greatest lower bound assigned by 𝒜 to any word in L. For boolean value sets, QWAs define boolean properties of traces, and QLAs define boolean properties of sets of traces, i.e., hyperproperties. For more general value sets, QLAs serve as a specification language for a generalization of hyperproperties, called quantitative hyperproperties. A nonprobabilistic (resp. probabilistic) quantitative hyperproperty assigns a value to each set (resp. distribution) G of traces, e.g., the minimal (resp. expected) average response time exhibited by the traces in G. We give several examples of quantitative hyperproperties and investigate three paradigmatic problems for QLAs: evaluation, nonemptiness, and universality. In the evaluation problem, given a QLA 𝔸 and an implementation G, we ask for the value that 𝔸 assigns to G. In the nonemptiness (resp. universality) problem, given a QLA 𝔸 and a value k, we ask whether 𝔸 assigns at least k to some (resp. every) language. We provide a comprehensive picture of decidability for these problems for QLAs with common aggregators as well as their restrictions to ω-regular languages and trace distributions generated by finite-state Markov chains.},
  author       = {Henzinger, Thomas A and Kebis, Pavol and Mazzocchi, Nicolas Adrien and Sarac, Naci E},
  booktitle    = {36th International Conference on Concurrency Theory},
  isbn         = {9783959773898},
  issn         = {1868-8969},
  location     = {Aarhus, Denmark},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Quantitative language automata}},
  doi          = {10.4230/LIPIcs.CONCUR.2025.21},
  volume       = {348},
  year         = {2025},
}

@inproceedings{20290,
  abstract     = {We consider equilibria in multiplayer stochastic graph games with terminal-node rewards. In such games, Nash equilibria are defined assuming that each player seeks to maximise their expected payoff, ignoring their aversion or tolerance to risk. We therefore study risk-sensitive equilibria (RSEs), where the expected payoff is replaced by a risk measure. A classical risk measure in the literature is the entropic risk measure, where each player has a real valued parameter capturing their risk-averseness. We introduce the extreme risk measure, which corresponds to extreme cases of entropic risk measure, where players are either extreme optimists or extreme pessimists. Under extreme risk measure, every player is an extremist: an extreme optimist perceives their reward as the maximum payoff that can be achieved with positive probability, while an extreme pessimist expects the minimum payoff achievable with positive probability. We argue that the extreme risk measure, especially in multi-player graph based settings, is particularly relevant as they can model several real life instances such as interactions between secure systems and potential security threats, or distributed controls for safety critical systems. We prove that RSEs defined with the extreme risk measure are guaranteed to exist when all rewards are non-negative. Furthermore, we prove that the problem of deciding whether a given game contains an RSE that generates risk measures within specified intervals is decidable and NP-complete for our extreme risk measure, and even PTIME-complete when all players are extreme optimists, while that same problem is undecidable using the entropic risk measure or even the classical expected payoff. This establishes, to our knowledge, the first decidable fragment for equilibria in simple stochastic games without restrictions on strategy types or number of players.},
  author       = {Brice, Léonard and Henzinger, Thomas A and Thejaswini, K. S.},
  booktitle    = {50th International Symposium on Mathematical Foundations of Computer Science},
  isbn         = {9783959773881},
  issn         = {1868-8969},
  location     = {Warsaw, Poland},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Finding equilibria: Simpler for pessimists, simplest for optimists}},
  doi          = {10.4230/LIPIcs.MFCS.2025.30},
  volume       = {345},
  year         = {2025},
}

@inproceedings{20291,
  abstract     = {We define and study classes of ω-regular automata for which the nondeterminism can be resolved by a policy that uses a combination of memory and randomness on any input word, based solely on the prefix read so far. We examine two settings for providing the input word to an automaton. In the first setting, called adversarial resolvability, the input word is constructed letter-by-letter by an adversary, dependent on the resolver’s previous decisions. In the second setting, called stochastic resolvability, the adversary pre-commits to an infinite word and reveals it letter-by-letter. In each setting, we require the existence of an almost-sure resolver, i.e., a policy that ensures that as long as the adversary provides a word in the language of the underlying nondeterministic automaton, the run constructed by the policy is accepting with probability 1.
The class of automata that are adversarially resolvable is the well-studied class of history-deterministic automata. The case of stochastically resolvable automata, on the other hand, defines a novel class. Restricting the class of resolvers in both settings to stochastic policies without memory introduces two additional new classes of automata. We show that the new automata classes offer interesting trade-offs between succinctness, expressivity, and computational complexity, providing a fine gradation between deterministic automata and nondeterministic automata.},
  author       = {Henzinger, Thomas A and Prakash, Aditya and Thejaswini, K. S.},
  booktitle    = {50th International Symposium on Mathematical Foundations of Computer Science},
  isbn         = {9783959773881},
  issn         = {1868-8969},
  location     = {Warsaw, Poland},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Resolving nondeterminism with randomness}},
  doi          = {10.4230/LIPIcs.MFCS.2025.57},
  volume       = {345},
  year         = {2025},
}

@inproceedings{20533,
  abstract     = {We give an introduction into differential privacy in the dynamic setting, called the continual observation setting.},
  author       = {Henzinger, Monika H and Safavi Hemami, Roodabeh},
  booktitle    = {33rd Annual European Symposium on Algorithms},
  isbn         = {9783959773959},
  issn         = {1868-8969},
  location     = {Warsaw, Poland},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Securing dynamic data: A primer on differentially private data structures}},
  doi          = {10.4230/LIPIcs.ESA.2025.2},
  volume       = {351},
  year         = {2025},
}

@inproceedings{20534,
  abstract     = {A non-trivial minimum cut (NMC) sparsifier is a multigraph Ĝ that preserves all non-trivial minimum cuts of a given undirected graph G. We introduce a flexible data structure for fully dynamic graphs that can efficiently provide an NMC sparsifier upon request at any point during the sequence of updates. We employ simple dynamic forest data structures to achieve a fast from-scratch construction of the sparsifier at query time. Based on the strength of the adversary and desired type of time bounds, the data structure comes with different guarantees. Specifically, let G be a fully dynamic simple graph with n vertices and minimum degree δ. Then our data structure supports an insertion/deletion of an edge to/from G in n^o(1) worst-case time. Furthermore, upon request, it can return w.h.p. an NMC sparsifier of G that has O(n/δ) vertices and O(n) edges, in Ô(n) time. The probabilistic guarantees hold against an adaptive adversary. Alternatively, the update and query times can be improved to Õ(1) and Õ(n) respectively, if amortized-time guarantees are sufficient, or if the adversary is oblivious. Throughout the paper, we use Õ to hide polylogarithmic factors and Ô to hide subpolynomial (i.e., n^o(1)) factors.
We discuss two applications of our new data structure. First, it can be used to efficiently report a cactus representation of all minimum cuts of a fully dynamic simple graph. Building this cactus for the NMC sparsifier instead of the original graph allows for a construction time that is sublinear in the number of edges. Against an adaptive adversary, we can with high probability output the cactus representation in worst-case Ô(n) time. Second, our data structure allows us to efficiently compute the maximal k-edge-connected subgraphs of undirected simple graphs, by repeatedly applying a minimum cut algorithm on the NMC sparsifier. Specifically, we can compute with high probability the maximal k-edge-connected subgraphs of a simple graph with n vertices and m edges in Õ(m+n²/k) time. This improves the best known time bounds for k = Ω(n^{1/8}) and naturally extends to the case of fully dynamic graphs.},
  author       = {Henzinger, Monika H and Kosinas, Evangelos and Münk, Robin and Räcke, Harald},
  booktitle    = {33rd Annual European Symposium on Algorithms},
  isbn         = {9783959773959},
  issn         = {1868-8969},
  location     = {Warsaw, Poland},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Efficient contractions of dynamic graphs - with applications}},
  doi          = {10.4230/LIPIcs.ESA.2025.36},
  volume       = {351},
  year         = {2025},
}

@inproceedings{20535,
  abstract     = {Many differentially private and classical non-private graph algorithms rely crucially on determining whether some property of each vertex meets a threshold. For example, for the k-core decomposition problem, the classic peeling algorithm iteratively removes a vertex if its induced degree falls below a threshold. The sparse vector technique (SVT) is generally used to transform non-private threshold queries into private ones with only a small additive loss in accuracy. However, a naive application of SVT in the graph setting leads to an amplification of the error by a factor of n due to composition, as SVT is applied to every vertex. In this paper, we resolve this problem by formulating a novel generalized sparse vector technique which we call the Multidimensional AboveThreshold (MAT) Mechanism which generalizes SVT (applied to vectors with one dimension) to vectors with multiple dimensions. When applied to vectors with n dimensions, we solve a number of important graph problems with better bounds than previous work.
Specifically, we apply our MAT mechanism to obtain a set of improved bounds for a variety of problems including k-core decomposition, densest subgraph, low out-degree ordering, and vertex coloring. We give a tight local edge differentially private (LEDP) algorithm for k-core decomposition that results in an approximation with O(ε^{-1} log n) additive error and no multiplicative error in O(n) rounds. We also give a new (2+η)-factor multiplicative, O(ε^{-1} log n) additive error algorithm in O(log² n) rounds for any constant η > 0. Both of these results are asymptotically tight against our new lower bound of Ω(log n) for any constant-factor approximation algorithm for k-core decomposition. Our new algorithms for k-core decomposition also directly lead to new algorithms for the related problems of densest subgraph and low out-degree ordering. Finally, we give novel LEDP differentially private defective coloring algorithms that use number of colors given in terms of the arboricity of the graph.},
  author       = {Dhulipala, Laxman and Henzinger, Monika H and Li, George Z. and Liu, Quanquan C. and Sricharan, A. R. and Zhu, Leqi},
  booktitle    = {33rd Annual European Symposium on Algorithms},
  isbn         = {9783959773959},
  issn         = {1868-8969},
  location     = {Warsaw, Poland},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Near-optimal differentially private graph algorithms via the Multidimensional AboveThreshold Mechanism}},
  doi          = {10.4230/LIPIcs.ESA.2025.91},
  volume       = {351},
  year         = {2025},
}

@inproceedings{20536,
  abstract     = {Uniquely represented (UR) data structures represent each logical state with a unique storage state. We study the problem of maintaining a dynamic set of n keys from a totally ordered universe in this context. UR structures are also called "strongly history independent" structures in the literature.
We introduce a two-layer data structure called (α,ε)-Randomized Block Search Tree (RBST) that is uniquely represented and suitable for external memory (EM). Though RBSTs naturally generalize the well-known binary Treaps, several new ideas are needed to analyze the expected search, update, and storage efficiency in terms of block-reads, block-writes, and blocks stored. We prove that searches have O(ε^{-1} + log_α n) block-reads, that dynamic updates perform O(ε^{-1} + log_α(n)/α) block-writes and O(ε^{-2}+(1+(ε^{-1}+log n)/α)log_α n) block-reads, and that (α, ε)-RBSTs have an asymptotic load-factor of at least (1-ε) for every ε ∈ (0,1/2].
Thus (α, ε)-RBSTs improve on the known, uniquely represented B-Treap [Golovin; ICALP'09]. Compared with non-UR structures, the RBST is also, to the best of our knowledge, the first external memory structure that is storage-efficient and has a non-amortized, write-efficient update bound.},
  author       = {Safavi Hemami, Roodabeh and Seybold, Martin P.},
  booktitle    = {19th International Symposium on Algorithms and Data Structures},
  isbn         = {9783959773980},
  issn         = {1868-8969},
  location     = {Toronto, Canada},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{B-Treaps revised: Write efficient randomized block search trees with high load}},
  doi          = {10.4230/LIPIcs.WADS.2025.47},
  volume       = {349},
  year         = {2025},
}

@inproceedings{20587,
  abstract     = {The blocks in the Bitcoin blockchain "record" the amount of work W that went into creating them through proofs of work. When honest parties control a majority of the work, consensus is achieved by picking the chain with the highest recorded weight. Resources other than work have been considered to secure such longest-chain blockchains. In Chia, blocks record the amount of disk-space S (via a proof of space) and sequential computational steps V (through a VDF).
In this paper, we ask what weight functions Γ(S,V,W) (that assign a weight to a block as a function of the recorded space, speed, and work) are secure in the sense that whenever the weight of the resources controlled by honest parties is larger than the weight of adversarial parties, the blockchain is secure against private double-spending attacks.
We completely classify such functions in an idealized "continuous" model: Γ(S,V,W) is secure against private double-spending attacks if and only if it is homogeneous of degree one in the "timed" resources V and W, i.e., αΓ(S,V,W) = Γ(S,α V, α W). This includes the Bitcoin rule Γ(S,V,W) = W and the Chia rule Γ(S,V,W) = S ⋅ V. In a more realistic model where blocks are created at discrete time-points, one additionally needs some mild assumptions on the dependency on S (basically, the weight should not grow too much if S is slightly increased, say linear as in Chia).
Our classification is more general and allows various instantiations of the same resource. It provides a powerful tool for designing new longest-chain blockchains. E.g., consider combining different PoWs to counter centralization, say the Bitcoin PoW W₁ and a memory-hard PoW W₂. Previous work suggested to use W₁+W₂ as weight. Our results show that using e.g., √{W₁}⋅ √{W₂} or min{W₁,W₂} are also secure, and we argue that in practice these are much better choices.},
  author       = {Baig, Mirza Ahad and Günther, Christoph Ullrich and Pietrzak, Krzysztof Z},
  booktitle    = {7th Conference on Advances in Financial Technologies},
  isbn         = {9783959774000},
  issn         = {1868-8969},
  location     = {Pittsburgh, PA, United States},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Nakamoto consensus from multiple resources}},
  doi          = {10.4230/LIPIcs.AFT.2025.16},
  volume       = {354},
  year         = {2025},
}

@inproceedings{21412,
  abstract     = {Payment channel networks (PCNs) are a promising technology that alleviates blockchain scalability by shifting the transaction load from the blockchain to the PCN. Nevertheless, the network topology has to be carefully designed to maximise the transaction throughput in PCNs. Additionally, users in PCNs also have to make optimal decisions on which transactions to forward and which to reject to prolong the lifetime of their channels. In this work, we consider an input sequence of transactions over p parties. Each transaction consists of a transaction size, source, and target, and can be either accepted or rejected (entailing a cost). The goal is to design a PCN topology among the p cooperating parties, along with the channel capacities, and then output a decision for each transaction in the sequence to minimise the cost of creating and augmenting channels, as well as the cost of rejecting transactions. Our main contribution is an 𝒪(p) approximation algorithm for the problem with p parties. We further show that with some assumptions on the distribution of transactions, we can reduce the approximation ratio to 𝒪(√p). We complement our theoretical analysis with an empirical study of our assumptions and approach in the context of the Lightning Network.},
  author       = {Chatterjee, Krishnendu and Křišťan, Jan Matyáš and Schmid, Stefan and Svoboda, Jakub and Yeo, Michelle X},
  booktitle    = {39th International Symposium on Distributed Computing},
  isbn         = {9783959774024},
  issn         = {1868-8969},
  location     = {Berlin, Germany},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Boosting payment channel network liquidity with topology optimization and transaction selection}},
  doi          = {10.4230/LIPIcs.DISC.2025.23},
  volume       = {356},
  year         = {2025},
}

@inproceedings{17144,
  abstract     = {We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.},
  author       = {Kourimska, Hana and Lieutier, André and Wintraecken, Mathijs},
  booktitle    = {40th International Symposium on Computational Geometry},
  isbn         = {9783959773164},
  issn         = {1868-8969},
  location     = {Athens, Greece},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms}},
  doi          = {10.4230/LIPIcs.SoCG.2024.69},
  volume       = {293},
  year         = {2024},
}

@inproceedings{17146,
  abstract     = {The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.},
  author       = {Edelsbrunner, Herbert and Pach, János},
  booktitle    = {40th International Symposium on Computational Geometry},
  isbn         = {9783959773164},
  issn         = {1868-8969},
  location     = {Athens, Greece},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Maximum Betti numbers of Čech complexes}},
  doi          = {10.4230/LIPIcs.SoCG.2024.53},
  volume       = {293},
  year         = {2024},
}

@inproceedings{17145,
  abstract     = {Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.},
  author       = {Rote, Günter and Rüber, Moritz and Saghafian, Morteza},
  booktitle    = {40th International Symposium on Computational Geometry},
  isbn         = {9783959773164},
  issn         = {1868-8969},
  location     = {Athens, Greece},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Grid peeling of parabolas}},
  doi          = {10.4230/LIPIcs.SoCG.2024.76},
  volume       = {293},
  year         = {2024},
}