We give faster algorithms for weak expander decompositions and approximate max flow on undirected graphs. First, we show that it is possible to "warm start" the cut-matching game when computing weak expander decompositions, avoiding the cost of the recursion depth. Our algorithm is also flexible enough to support weaker flow subroutines than previous algorithms. Our second contribution is to streamline the recent non-recursive approximate max flow algorithm of Li, Rao, and Wang (SODA, 2025) and adapt their framework to use our new weak expander decomposition primitive. Consequently, we give an approximate max flow algorithm within a few logarithmic factors of the limit of expander decomposition-based approaches.

We aim to revive Thorup’s conjecture [Thorup, WG’92] on the existence of reachability shortcuts with ideal size-diameter tradeoffs. Thorup originally asked whether, given any graph G=(V,E) with m edges, we can add m1+o(1) “shortcut” edges E+ from the transitive closure E* of G so that G+(u,v) ≤ mo(1) for all (u,v)∈ E*, where G+=(V,E∪ E+). The conjecture was refuted by Hesse [Hesse, SODA’03], followed by significant efforts in the last few years to optimize the lower bounds.

Abstract We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in ([1]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms ([2, 3]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read- k families [4] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [4].

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.

We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of $\tilde{O}(nm)$ for weighted graphs and $\tilde{O}\left(\mathrm{~m}^{3/2}\right)$ for unweighted graphs (Gabow 1995) for this problem. The running time of our algorithm is dominated by a logarithmic number of calls to a directed global minimum cut subroutine – if the running time of the latter problem improves to $m^{1+o(1)}$ (thereby matching the running time of maximum flow), the running time of our arboricity algorithm would improve further to $m^{1+o(1)}$. We also give a new algorithm for computing the entire cut hierarchy – laminar multiway cuts with minimum cut ratio in recursively defined induced subgraphs – in $m n^{1+o(1)}$ time. The cut hierarchy yields the ideal edge loads (Thorup 2001) in a fractional spanning tree packing of the graph which, we show, also corresponds to a max-entropy solution in the spanning tree polytope. For the cut hierarchy problem, the previous best bound was $\tilde{O}\left(n^{2} m\right)$ for weighted graphs and $\tilde{O}\left(n m^{3/2}\right)$ for unweighted graphs.

We study vertex sparsification for preserving distances in planar graphs. Given an edge-weighted planar graph with k terminals, the goal is to construct an emulator, which is a smaller edge-weighted planar graph that contains the terminals and exactly preserves the pairwise distances between them. We construct exact planar emulators of size $O\left(f^{2} k^{2}\right)$ in the setting where terminals lie on f faces in the planar embedding of the input graph. Our result generalizes and interpolates between the previous results of Chang and Ophelders and Goranci, Henzinger, and Peng which is an $O\left(k^{2}\right)$ bound in the setting where all terminals lie on a single face (i.e., f = 1), and the result of Krauthgamer, Nguyen, and Zondiner, which is an $O\left(k^{4}\right)$ bound for the general case (i.e., f=k).Our construction follows a recent new way of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

Recently differential privacy has been used for a number of streaming, data structure, and dynamic graph problems as a means of hiding the internal randomness of the data structure, so that multiple possibly adaptive queries can be made without sacrificing the correctness of the responses. Although these works use differential privacy to show that for some problems it is possible to tolerate $T$ queries using $\widetilde{O}(\sqrt{T})$ copies of a data structure, such results only apply to numerical estimation problems, and only return the cost of an optimization problem rather than the solution itself. In this paper we investigate the use of differential privacy for adaptive queries to search problems, which are significantly more challenging since the responses to queries can reveal much more about the internal randomness than a single numerical query. We focus on two classical search problems: nearest neighbor queries and regression with arbitrary turnstile updates. We identify key parameters to these problems, such as the number of $c$-approximate near neighbors and the matrix condition number, and use different differential privacy techniques to design algorithms returning the solution point or solution vector with memory and time depending on these parameters. We give algorithms for each of these problems that achieve similar tradeoffs.

We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in ([Frederick Qiu and Sahil Singla, 2022]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms ([Chandra Chekuri et al., 2010; Nicholas J. A. Harvey and Neil Olver, 2014]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [Dmitry Gavinsky et al., 2015] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [Dmitry Gavinsky et al., 2015].

Vaccines have proven to be extremely effective in preventing the spread of COVID-19 and potentially ending the pandemic. Lack of access caused many people not getting vaccinated early, so states such as Virginia deployed mobile vaccination sites in order to distribute vaccines across the state. Here we study the problem of deciding where these facilities should be placed and moved over time in order to minimize the distance each person needs to travel in order to be vaccinated. Traditional facility location models for this problem fail to incorporate the fact that our facilities are mobile (i.e., they can move over time). To this end, we instead model vaccine distribution as the Dynamic k-Supplier problem and give the first approximation algorithms for this problem. We then run extensive simulations on real world datasets to show the efficacy of our methods. In particular, we find that natural baselines for Dynamic k-Supplier cannot take advantage of the mobility of the facilities, and perform worse than non-mobile k-Supplier algorithms.

Set Cover is a fundamental problem in combinatorial optimization which has been studied for many decades due to its various applications across multiple domains. In many of these domains, the input data consists of locations, relationships, and other sensitive information of individuals which may leaked due to the set cover output. Attempts have been made to design privacy-preserving algorithms to solve the Set Cover under privacy constraints. Under differential privacy, it has been proved that the Set Cover problem has strong impossibility results and no explicit forms of the output can be released to the public. In this work, we observe that these hardness results dissolve when we turn to the Partial Set Cover problem, where we only need to cover a ρ ∈ (0,1) fraction of the elements. We show that this relaxation enables us to avoid the impossibility results, and give the first algorithm which outputs an explicit form of set cover with non-trivial utility guarantees under differential privacy. Using our algorithm as a subroutine, we design a differentially private bicriteria algorithm to solve a recently proposed facility location problem for vaccine distribution which generalizes the k-supplier with outliers. Our analysis shows that relaxing the covering requirement to serve only a ρ ∈ (0,1) fraction of the population/universe also allows us to circumvent the inherent hardness of k-supplier and give the first non-trivial guarantees.

Multi-item revenue-optimal mechanisms are known to be extremely complex, often offering buyers randomized lotteries of goods. In the standard buy-one model, it is known that optimal mechanisms can yield revenue infinitely higher than that of any "simple" mechanism---the ones with size polynomial in the number of items---even with just two items and a single buyer [Briest et al. 2015; Hart and Nisan 2017].