@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{19982, abstract = {Dynamically maintaining the minimum cut in a graph G under edge insertions and deletion is a fundamental problem in dynamic graph algorithms for which no conditional lower bound on the time per operation exists. In an n-node graph the best known (1 + o (1))-approximate algorithm takes update time [14]. If the minimum cut is guaranteed to be (log n )o (1), a deterministic exact algorithm with n o (1) update time exists [8]. We present the first fully dynamic algorithm for (1 + o (1))-approximate minimum cut with n o(1) update time. Our main technical contribution is to show that it suffices to consider small-volume cuts in suitably contracted graphs.}, author = {El-Hayek, Antoine and Henzinger, Monika H and Li, Jason}, booktitle = {Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms}, location = {New Orleans, LA, United States}, pages = {750--784}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Fully dynamic approximate minimum cut in subpolynomial time per operation}}, doi = {10.1137/1.9781611978322.22}, year = {2025}, } @inproceedings{20051, abstract = {We revisit the majority problem in the population protocol communication model, as first studied by Angluin et al. (Distributed Computing 2008). We consider a more general version of this problem known as plurality consensus, which has already been studied intensively in the literature. In this problem, each node in a system of n nodes, has initially one of k different opinions, and they need to agree on the (relative) majority opinion. In particular, we consider the important and intensively studied model of Undecided State Dynamics. Our main contribution is an almost tight lower bound on the stabilization time: we prove that there exists an initial configuration, even with bias \Delta = \omega(\sqrt{n\log n}), where stabilization requires \Omega(kn\log \frac {\sqrt n} {k \log n}) interactions, or equivalently, \Omega(k\log \frac {\sqrt n} {k \log n}) parallel time for any k = o\left(\frac {\sqrt n}{\log n}\right). This bound is tight for any k \le n^{\frac 1 2 - \epsilon}, where \epsilon >0 can be any small constant, as Amir et al.~(PODC'23) gave a O(k\log n) parallel time upper bound for k = O\left(\frac {\sqrt n} {\log ^2 n}\right).}, author = {El-Hayek, Antoine and Elsässer, Robert and Schmid, Stefan}, booktitle = {Proceedings of the ACM Symposium on Principles of Distributed Computing}, isbn = { 9798400718854}, location = {Huatulco, Mexico}, publisher = {Association for Computing Machinery}, title = {{An almost tight lower bound for plurality consensus with undecided state dynamics in the population protocol model}}, doi = {10.1145/3732772.3733505}, year = {2025}, } @inproceedings{20052, abstract = {This paper revisits a fundamental distributed computing problem in the population protocol model. Provided n agents each starting with an input color in [k], the relative majority problem asks to find the predominant color. In the population protocol model, at each time step, a scheduler selects two agents that first learn each other's states and then update their states based on what they learned. We present the Circles protocol that solves the relative majority problem with k3 states. It is always-correct under weakly fair scheduling. Not only does it improve upon the best known upper bound of O(k7), but it also shows a strikingly simpler design inspired by energy minimization in chemical settings.}, author = {Breitkopf, Tom-Lukas and Dallot, Julien and El-Hayek, Antoine and Schmid, Stefan}, booktitle = {Proceedings of the ACM Symposium on Principles of Distributed Computing}, isbn = {9798400718854}, location = {Huatulco, Mexico}, pages = {549--552}, publisher = {Association for Computing Machinery}, title = {{Brief announcement: Minimizing energy solves relative majority with a cubic number of states in population protocols}}, doi = {10.1145/3732772.3733512}, year = {2025}, } @inproceedings{20301, abstract = {We study privately releasing column sums of a d-dimensional table with entries from a universe χ undergoing T row updates, called histogram under continual release. Our mechanisms give better additive ℓ∞-error than existing mechanisms for a large class of queries and input streams. Our first contribution is an output-sensitive mechanism in the insertions-only model (χ = {0, 1}) for maintaining (i) the histogram or (ii) queries that do not require maintaining the entire histogram, such as the maximum or minimum column sum, the median, or any quantiles. The mechanism has an additive error of O(d log2 (dq∗) + log T) whp, where q∗ is the maximum output value over all time steps on this dataset. The mechanism does not require q∗ as input. This breaks the Ω(d log T) bound of prior work when q∗ ≪ T. Our second contribution is a mechanism for the turnstile model that admits negative entry updates (χ = {−1, 0, 1}). This mechanism has an additive error of O(d log2(dK) + log T) whp, where K is the number of times two consecutive data rows differ, and the mechanism does not require K as input. This is useful when monitoring inputs that only vary under unusual circumstances. For d = 1 this gives the first private mechanism with error O(log2 K + log T) for continual counting in the turnstile model, improving on the O(log2 n + log T) error bound by Dwork et al. (2015), where n is the number of ones in the stream, as well as allowing negative entries, while Dwork et al. (2015) can only handle nonnegative entries (χ = {0, 1}). }, author = {Henzinger, Monika H and Sricharan, A. R. and Steiner, Teresa Anna}, booktitle = {The 28th International Conference on Artificial Intelligence and Statistics}, issn = {2640-3498}, location = {Mai Khao, Thailand}, pages = {1990--1998}, publisher = {ML Research Press}, title = {{Differentially private continual release of histograms and related queries}}, volume = {258}, 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{19038, abstract = {Differentially private weighted prefix sum under continual observation is a crucial component in the production-level deployment of private next-word prediction for Gboard, which, according to Google, has over a billion users. More specifically, Google uses a differentially private mechanism to sum weighted gradients in its private follow-the-regularized leader algorithm. Apart from efficiency, the additive error of the private mechanism is crucial as multiplied with the square root of the model’s dimension d (with d ranging up to 10 trillion, for example, Switch Transformers or M6-10T), it determines the accuracy of the learning system. So, any improvement in leading constant matters significantly in practice. In this paper, we show a novel connection between mechanisms for continual weighted prefix sum and a concept in representation theory known as the group matrix introduced in correspondence between Dedekind and Frobenius (Sitzungsber. Preuss. Akad. Wiss. Berlin, 1897) and generalized by Schur (Journal für die reine und angewandte Mathematik, 1904). To the best of our knowledge, this is the first application of group algebra in the analysis of differentially private algorithms. Using this connection, we analyze a class of matrix norms known as factorization norms that give upper and lower bounds for the additive error under general ℓp-norms of the matrix mechanism. This allows us to give 1. the first efficient factorization that matches the best-known non-constructive upper bound on the factorization norm by Mathias (SIAM Journal of Matrix Analysis and Applications, 1993) for the matrix used in Google’s deployment, and also improves on the previous best-known constructive bound of Fichtenberger, Henzinger, and Upadhyay (ICML 2023) and Henzinger, Upadhyay, and Upadhyay (SODA 2023); thereby, partially resolving an open question in operator theory, 2. the first upper bound on the additive error for a large class of weight functions for weighted prefix sum problems, including the sliding window matrix (Bolot, Fawaz, Muthukrishnan, Nikolov, and Taft (ICDT 2013). We also improve the bound on factorizing the striped matrix used for outputting a synthetic graph that approximates all cuts (Fichtenberger, Henzinger, and Upadhyay (ICML 2023)); 3. a general improved upper bound on the factorization norms that depend on algebraic properties of the weighted sum matrices and that applies to a more general class of weighting functions than the ones considered in Henzinger, Upadhyay, and Upadhyay (SODA 2024). Using the known connection between these factorization norms and the ℓp-error of continual weighted sum, we give an upper bound on the ℓp-error for the continual weighted sum problem for p ≥ 2.}, author = {Henzinger, Monika H and Upadhyay, Jalaj}, booktitle = {Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms}, isbn = {979-833131200-8}, issn = {1071-9040}, location = {New Orleans, LA, United States}, pages = {2951 -- 2970}, publisher = {Association for Computing Machinery}, title = {{Improved differentially private continual observation using group algebra}}, doi = {10.1137/1.9781611978322.95}, volume = {5}, year = {2025}, } @article{15121, abstract = {We present an auction algorithm using multiplicative instead of constant weight updates to compute a (1-E)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time 0(mE-1), beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM ’14] that runs in 0(mE-1 log E-1). Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a (1-E)-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is 0(mE-1), where m is the sum of the number of initially existing and inserted edges.}, author = {Zheng, Da Wei and Henzinger, Monika H}, issn = {1436-4646}, journal = {Mathematical Programming}, pages = {881--894}, publisher = {Springer Nature}, title = {{Multiplicative auction algorithm for approximate maximum weight bipartite matching}}, doi = {10.1007/s10107-024-02066-3}, volume = {210}, year = {2025}, } @inproceedings{21280, abstract = {We give an algorithm that, with high probability, maintains a (1-ε)-approximate s-t maximum flow in undirected, uncapacitated n-vertex graphs undergoing m edge insertions in Õ(m+ n F^*/ε) total update time, where F^{*} is the maximum flow on the final graph. This is the first algorithm to achieve polylogarithmic amortized update time for dense graphs (m = Ω(n²)), and more generally, for graphs where F^* = Õ(m/n). At the heart of our incremental algorithm is the residual graph sparsification technique of Karger and Levine [SICOMP '15], originally designed for computing exact maximum flows in the static setting. Our main contributions are (i) showing how to maintain such sparsifiers for approximate maximum flows in the incremental setting and (ii) generalizing the cut sparsification framework of Fung et al. [SICOMP '19] from undirected graphs to balanced directed graphs.}, author = {Goranci, Gramoz and Henzinger, Monika H and Räcke, Harald and Sricharan, A.}, booktitle = {52nd International Colloquium on Automata, Languages, and Programming}, isbn = {9783959773720}, location = {Aarhus, Denmark}, pages = {91:1--91:20}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Incremental approximate maximum flow via residual graph sparsification}}, doi = {10.4230/lipics.icalp.2025.91}, volume = {334}, year = {2025}, } @inproceedings{18115, abstract = {We study the data selection problem, whose aim is to select a small representative subset of data that can be used to efficiently train a machine learning model. We present a new data selection approach based on k-means clustering and sensitivity sampling. Assuming access to an embedding representation of the data with respect to which the model loss is Holder continuous, our approach provably allows selecting a set of “typical” k+1/ε2 elements whose average loss corresponds to the average loss of the whole dataset, up to a multiplicative (1±ε) factor and an additive ελΦk, where Φk represents the k-means cost for the input embeddings and λ is the Holder constant. We furthermore demonstrate the performance and scalability of our approach on fine-tuning foundation models and show that it outperforms state-of-the-art methods. We also show how it can be applied on linear regression, leading to a new sampling strategy that surprisingly matches the performance of leverage score sampling, while being conceptually simpler and more scalable.}, author = {Axiotis, Kyriakos and Cohen-Addad, Vincent and Henzinger, Monika H and Jerome, Sammy and Mirrokni, Vahab and Saulpic, David and Woodruff, David P. and Wunder, Michael}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, issn = {2640-3498}, location = {Vienna, Austria}, pages = {2086--2107}, publisher = {ML Research Press}, title = {{Data-efficient learning via clustering-based sensitivity sampling: Foundation models and beyond}}, volume = {235}, year = {2024}, } @inproceedings{18116, abstract = {As a staple of data analysis and unsupervised learning, the problem of private clustering has been widely studied, under various privacy models. Centralized differential privacy is the first of them, and the problem has also been studied for the local and the shuffle variation. In each case, the goal is to design an algorithm that computes privately a clustering, with the smallest possible error. The study of each variation gave rise to new algorithm: the landscape of private clustering algorithm is therefore quite intricate. In this paper, we show that a 20 year-old algorithm can be slightly modified to work for any of those models. This provides a unified picture: while matching almost all previously known results, it allows us to improve some of them, and extend to a new privacy model, the continual observation setting, where the input is changing over time and the algorithm must output a new solution at each time step.}, author = {La Tour, Max Dupré and Henzinger, Monika H and Saulpic, David}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, issn = {2640-3498}, location = {Vienna, Austria}, pages = {12046--12086}, publisher = {ML Research Press}, title = {{Making old things new: A unified algorithm for differentially private clustering}}, volume = {235}, year = {2024}, } @inproceedings{18156, abstract = {Privately counting distinct elements in a stream is a fundamental data analysis problem with many applications in machine learning. In the turnstile model, Jain et al. [NeurIPS2023] initiated the study of this problem parameterized by the maximum flippancy of any element, i.e., the number of times that the count of an element changes from 0 to above 0 or vice versa. They give an item-level (ε,δ)-differentially private algorithm whose additive error is tight with respect to that parameterization. In this work, we show that a very simple algorithm based on the sparse vector technique achieves a tight additive error for item-level (ε,δ)-differential privacy and item-level ε-differential privacy with regards to a different parameterization, namely the sum of all flippancies. Our second result is a bound which shows that for a large class of algorithms, including all existing differentially private algorithms for this problem, the lower bound from item-level differential privacy extends to event-level differential privacy. This partially answers an open question by Jain et al. [NeurIPS2023].}, author = {Henzinger, Monika H and Sricharan, A. R. and Steiner, Teresa Anna}, booktitle = {International Conference on Approximation Algorithms for Combinatorial Optimization Problems }, isbn = {9783959773485}, issn = {1868-8969}, location = {London, United Kingdom}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Private counting of distinct elements in the turnstile model and extensions}}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.40}, volume = {317}, year = {2024}, } @inproceedings{18308, abstract = {We study in this paper the problem of maintaining a solution to k-median and k-means clustering in a fully dynamic setting. To do so, we present an algorithm to efficiently maintain a coreset, a compressed version of the dataset, that allows easy computation of a clustering solution at query time. Our coreset algorithm has near-optimal update time of Õ(k) in general metric spaces, which reduces to Õ(d) in the Euclidean space ℝ^d. The query time is O(k²) in general metrics, and O(kd) in ℝ^d. To maintain a constant-factor approximation for k-median and k-means clustering in Euclidean space, this directly leads to an algorithm with update time Õ(d), and query time Õ(kd + k²). To maintain a O(polylog k)-approximation, the query time is reduced to Õ(kd).}, author = {La Tour, Max Dupré and Henzinger, Monika H and Saulpic, David}, booktitle = {32nd Annual European Symposium on Algorithms}, isbn = {9783959773386}, issn = {1868-8969}, location = {London, United Kingdom}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Fully dynamic k-means coreset in near-optimal update time}}, doi = {10.4230/LIPIcs.ESA.2024.100}, volume = {308}, year = {2024}, } @inproceedings{18503, abstract = {In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time O(m log3 n) which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs. Previously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs (which was improved to have running time O(m log2 n log log n) in [HRW20].) The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an m1+o(1) deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed “almost-linear”. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, “derandomize” the methods of Karger. In terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with o(1) error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to 1 + o(1/ log n)-approximate so that they can be applied recursively as in [Li21] over O(log n) many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters.}, author = {Henzinger, Monika H and Li, Jason and Rao, Satish and Wang, Di}, booktitle = {35th Annual ACM-SIAM Symposium on Discrete Algorithms}, location = {Alexandria, VA, United States}, pages = {3089--3139}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Deterministic near-linear time minimum cut in weighted graphs}}, doi = {10.1137/1.9781611977912.111}, year = {2024}, } @inproceedings{18557, abstract = {Broadcast and Consensus are most fundamental tasks in distributed computing. These tasks are particularly challenging in dynamic networks where communication across the network links may be unreliable, e.g., due to mobility or failures. Over the last years, researchers have derived several impossibility results and high time complexity lower bounds for these tasks. Specifically for the setting where in each round of communication the adversary is allowed to choose one rooted tree along which the information is disseminated, there is a lower as well as an upper bound that is linear in the number n of nodes for Broadcast and for n ≥ 3 the adversary can guarantee that Consensus never happens. This setting is called the oblivious message adversary for rooted trees. Also note that if the adversary is allowed to choose a graph that does not contain a rooted tree, then it can guarantee that Broadcast and Consensus will never happen. However, such deterministic adversarial models may be overly pessimistic, as many processes in real-world settings are stochastic in nature rather than worst-case. This paper studies Broadcast on stochastic dynamic networks and shows that the situation is very different to the deterministic case. In particular, we show that if information dissemination occurs along random rooted trees and directed Erdős–Rényi graphs, Broadcast completes in O(log n) rounds of communication with high probability. The fundamental insight in our analysis is that key variables are mutually independent. We then study two adversarial models, (a) one with Byzantine nodes and (b) one where an adversary controls the edges. (a) Our techniques without Byzantine nodes are general enough so that they can be extended to Byzantine nodes. (b) In the spirit of smoothed analysis, we introduce the notion of randomized oblivious message adversary, where in each round, an adversary picks k ≤ 2n/3 edges to appear in the communication network, and then a graph (e.g. rooted tree or directed Erdős–Rényi graph) is chosen uniformly at random among the set of all such graphs that include these edges. We show that Broadcast completes in a finite number of rounds, which is, e.g., O(k+log n) rounds in rooted trees. We then extend these results to All-to-All Broadcast, and Consensus, and give lower bounds that show that most of our upper bounds are tight.}, author = {El-Hayek, Antoine and Henzinger, Monika H and Schmid, Stefan}, booktitle = {38th International Symposium on Distributed Computing}, isbn = {9783959773522}, issn = {1868-8969}, location = {Madrid, Spain}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Broadcast and Consensus in stochastic dynamic networks with Byzantine nodes and adversarial edges}}, doi = {10.4230/LIPIcs.DISC.2024.21}, volume = {319}, year = {2024}, } @inproceedings{18906, abstract = {Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their inherent intricacies and large hidden factors in their asymptotic running times. Here, we introduce the first practically efficient algorithm for computing expander decompositions and their hierarchies and demonstrate its effectiveness and utility by incorporating it as the core component in a novel solver for the normalized cut graph clustering objective. Our extensive experiments on a variety of large graphs show that our expander-based algorithm outperforms state-of-the-art solvers for normalized cut with respect to solution quality by a large margin on a variety of graph classes such as citation, e-mail, and social networks or web graphs while remaining competitive in running time.}, author = {Hanauer, Kathrin and Henzinger, Monika H and Münk, Robin and Räcke, Harald and Vötsch, Maximilian}, booktitle = {Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining}, isbn = {9798400704901}, location = {Barcelona, Spain}, pages = {1016--1027}, publisher = {ACM}, title = {{Expander hierarchies for normalized cuts on graphs}}, doi = {10.1145/3637528.3671978}, year = {2024}, } @inproceedings{18928, abstract = {Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update. We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate. Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight. We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds.}, author = {Henzinger, Monika H and Saha, Barna and Seybold, Martin P. and Ye, Christopher}, booktitle = {15th Innovations in Theoretical Computer Science Conference}, isbn = {9783959773096}, issn = {1868-8969}, location = {Berkeley, CA, United States}, pages = {62:1--62:25}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{On the complexity of algorithms with predictions for dynamic graph problems}}, doi = {10.4230/LIPIcs.ITCS.2024.62}, volume = {287}, year = {2024}, } @inproceedings{19512, abstract = {Differential privacy with gradual expiration models the setting where data items arrive in a stream and at a given time t the privacy loss guaranteed for a data item seen at time (t − d) is εg(d), where g is a monotonically non-decreasing function. We study the fundamental continual (binary) counting problem where each data item consists of a bit, and the algorithm needs to output at each time step the sum of all the bits streamed so far. For a stream of length T and privacy without expiration continual counting is possible with maximum (over all time steps) additive error O(log2 (T)/ε) and the best known lower bound is Ω(log(T)/ε); closing this gap is a challenging open problem. We show that the situation is very different for privacy with gradual expiration by giving upper and lower bounds for a large set of expiration functions g. Specifically, our algorithm achieves an additive error of O(log(T)/ε) for a large set of privacy expiration functions. We also give a lower bound that shows that if C is the additive error of any ε-DP algorithm for this problem, then the product of C and the privacy expiration function after 2C steps must be Ω(log(T)/ε). Our algorithm matches this lower bound as its additive error is O(log(T)/ε), even when g(2C) = O(1). Our empirical evaluation shows that we achieve a slowly growing privacy loss with significantly smaller empirical privacy loss for large values of d than a natural baseline algorithm.}, author = {Andersson, Joel Daniel and Henzinger, Monika H and Pagh, Rasmus and Steiner, Teresa Anna and Upadhyay, Jalaj}, booktitle = {38th Conference on Neural Information Processing Systems}, issn = {1049-5258}, location = {Vancouver, Canada}, publisher = {Neural Information Processing Systems Foundation}, title = {{Continual counting with gradual privacy expiration}}, volume = {37}, year = {2024}, }