arXiv subjects

George Z. Li

Publications and source records attributed to George Z. Li.

18 recordsLinked to original sources

arXiv2026

The Dirichlet Mechanism for rounding with strong negative correlation, with applications

Many optimization and scheduling problems can be abstracted in terms of a bipartite ``assignment graph" $G = (U \cup V, E)$, where the goal is to select exactly one edge for each right-node. For example, a right-node may correspond to a job, and a left-node to a possible machine assignment. A common strategy to solve such problems is to obtain a fractional relaxation $x_e$ for each edge $e$, and then have each right-node independently select an edge with probability $x_e$. However, this may cause the left-nodes to become unevenly loaded, leading to suboptimal solutions for some problems. To address this, a number of algorithms for dependent rounding with strong negative correlation have been developed, e.g. Bansal, Srinivasan & Svensson (2021), Im & Shadloo (2020), Im & Li (2023), Harris (2024), Naor, Srinivasan & Wajc (2025). We introduce a new method for this, which we call the \emph{Dirichlet mechanism}. It is based on having each left-node draw Dirichlet random variables for its edges, and then having each right-node select an edge based on these values. This achieves quantitatively stronger negative correlation than previous algorithms, and is also simpler since it avoids the need for a tie-breaking mechanism. We illustrate the mechanism with improved approximation ratios for two problems. For oblivious online dependent rounding, we achieve a $0.68$-approximation which improves upon the previous $0.652$-approximation of Naor, Srinivasan & Wajc (2025). For the problem of scheduling jobs on unrelated machines to minimize weighted completion time, we achieve a $1.387$-approximation which improves upon the $1.398$-approximation of Harris (2024). (A recent algorithm of Li (2025) based on iterated rounding also provides a $1.36$-approximation if the weights of each job are independent of machine.)

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arXiv2026

Improved Strongly Polynomial Work-Span Tradeoffs for Directed Single Source Shortest Paths

We revisit the single-source shortest paths (SSSP) problem on directed graphs with nonnegative real weights and give a deterministic parallel algorithm with $O(n^{1+o(1)}t^2 + m^{1+o(1)})$ work and $\tilde{O}(n/t)$ span, for any $t \in [1, n]$. This matches (up to subpolynomial factors) the tradeoff due to [Shi and Spencer '99] for undirected graphs with nonnegative real weights.

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arXiv2026

Bellman-Ford in Almost-Linear Time

We consider the single-source shortest paths problem on a directed graph with real-valued (possibly negative) edge weights and solve this problem in $m^{1+o(1)}$ time.

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arXiv2026

Bellman-Ford in Almost-Linear Time for Dense Graphs

We consider the single-source shortest paths problem on a directed graph with real-valued (possibly negative) edge weights and solve this problem in $n^{2+o(1)}$ time by refining the shortcutting procedure introduced in Li, Li, Rao, and Zhang (2026).

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arXiv2026

Differentially Private Matchings

Computing matchings in graphs is a foundational algorithmic task. Despite extensive interest in differentially private (DP) graph analysis, work on privately computing matching solutions, rather than just their size, has been sparse. The sole prior work in the standard model of pure $\varepsilon$-differential privacy, by Hsu, Huang, Roth, Roughgarden, and Wu [HHR+14, STOC'14], focused on allocations and was thus restricted to bipartite graphs. We present a comprehensive study of DP algorithms for maximum matching and b-matching in general graphs, which also yields techniques that improve upon the bipartite setting. En route to solving these matching problems, we develop a set of novel techniques with broad applicability, including a new symmetry argument for DP lower bounds, the first arboricity-based sparsifiers for node-DP, and the novel Public Vertex Subset Mechanism.

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arXiv2025

Shortcutting for Negative-Weight Shortest Path

Consider the single-source shortest paths problem on a directed graph with real-valued edge weights. We solve this problem in $O(n^{2.5}\log^{4.5}n)$ time, improving on prior work of Fineman (STOC 2024) and Huang-Jin-Quanrud (SODA 2025, 2026) on dense graphs. Our main technique is an shortcutting procedure that iteratively reduces the number of negative-weight edges along shortest paths by a constant factor.

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arXiv2025

Improved Directed Expander Decompositions

We obtain faster expander decomposition algorithms for directed graphs, matching the guarantees of Saranurak and Wang (SODA 2019) for expander decomposition on undirected graphs. Our algorithms are faster than prior work and also generalize almost losslessly to capacitated graphs. In particular, we obtain the first directed expander decomposition algorithm for capacitated graphs in near-linear time with optimal dependence on $ϕ$. To obtain our result, we provide the first implementation and analysis of the non-stop cut-matching game for directed, capacitated graphs. All existing directed expander decomposition algorithms instead temporarily add ''fake edges'' before pruning them away in a final cleanup step. Our result shows that the natural undirected approach applies even to directed graphs. The difficulty is in its analysis, which is technical and requires significant modifications from the original setting of undirected graphs.

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arXiv2025

Faster Weak Expander Decompositions and Approximate Max Flow

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.

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arXiv2025

Reviving Thorup's Shortcut Conjecture

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 $m^{1+o(1)}$ ``shortcut'' edges $E_+$ from the transitive closure $E^*$ of $G$ so that $\text{dist}_{G_+}(u,v) \leq m^{o(1)}$ for all $(u,v)\in E^*$, where $G_+=(V,E\cup E_+)$. The conjecture was refuted by Hesse [Hesse, SODA'03], followed by significant efforts in the last few years to optimize the lower bounds. In this paper we observe that although Hesse refuted the letter of Thorup's conjecture, his work~[Hesse, SODA'03] -- and all followup work -- does not refute the spirit of the conjecture, which should allow $G_+$ to contain both new (shortcut) edges and new Steiner vertices. Our results are as follows. (1) On the positive side, we present explicit attacks that break all known shortcut lower bounds when Steiner vertices are allowed. (2) On the negative side, we rule out ideal $m^{1+o(1)}$-size, $m^{o(1)}$-diameter shortcuts whose ``thickness'' is $t=o(\log n/\log \log n)$, meaning no path can contain $t$ consecutive Steiner vertices. (3) We propose a candidate hard instance as the next step toward resolving the revised version of Thorup's conjecture. Finally, we show promising implications. Almost-optimal parallel algorithms for computing a generalization of the shortcut that approximately preserves distances or flows imply almost-optimal parallel algorithms with $m^{o(1)}$ depth for exact shortcut paths and exact maximum flow. The state-of-the-art algorithms have much worse depth of $n^{1/2+o(1)}$ [Rozhoň, Haeupler, Martinsson, STOC'23] and $m^{1+o(1)}$ [Chen, Kyng, Liu, FOCS'22], respectively.

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arXiv2025

Near-Optimal Differentially Private Graph Algorithms via the Multidimensional AboveThreshold Mechanism

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. As an application, we solve a number of important graph problems with better bounds than previous work. 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 DP algorithm for $k$-core decomposition 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 $O(\log^2 n)$ 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 also directly lead to new algorithms for densest subgraph and low out-degree ordering. Our novel private defective coloring algorithms uses number of colors proportional to the arboricity of the graph.

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arXiv2025

Fast Algorithms for Graph Arboricity and Related Problems

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}(m^{3/2}) $ 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 $\tilde{O}(n^2 m)$ for weighted graphs and $\tilde{O}(n m^{3/2})$ for unweighted graphs.

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arXiv2025

Paths and Intersections: Exact Emulators for Planar 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 $O(f^2k^2)$ 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 $O(k^2)$ 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 $O(k^4)$ 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.

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arXiv2025

On Differential Privacy for Adaptively Solving Search Problems via Sketching

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 vector with memory and time depending on these parameters. We give algorithms for each of these problems that achieve similar tradeoffs.

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arXiv2024

Concentration of Submodular Functions and Read-k Families Under Negative Dependence

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 (Qiu and Singla [QS22]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms (Chekuri, Vondrak, and Zenklusen [CVZ10] and Harvey and Olver [HO14]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [Gav+15] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [Gav+15].

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arXiv2024

Fine-Grained Privacy Guarantees for Coverage Problems

We introduce a new notion of neighboring databases for coverage problems such as Max Cover and Set Cover under differential privacy. In contrast to the standard privacy notion for these problems, which is analogous to node-privacy in graphs, our new definition gives a more fine-grained privacy guarantee, which is analogous to edge-privacy. We illustrate several scenarios of Set Cover and Max Cover where our privacy notion is desired one for the application. Our main result is an $ε$-edge differentially private algorithm for Max Cover which obtains an $(1-1/e-η,\tilde{O}(k/ε))$-approximation with high probability. Furthermore, we show that this result is nearly tight: we give a lower bound show that an additive error of $Ω(k/ε)$ is necessary under edge-differential privacy. Via group privacy properties, this implies a new algorithm for $ε$-node differentially private Max Cover which obtains an $(1-1/e-η,\tilde{O}(fk/ε))$-approximation, where $f$ is the maximum degree of an element in the set system. When $f\ll k$, this improves over the best known algorithm for Max Cover under pure (node) differential privacy, which obtains an $(1-1/e,\tilde{O}(k^2/ε))$-approximation.

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arXiv2024

Near-Optimal Differentially Private k-Core Decomposition

Recent work by Dhulipala et al. \cite{DLRSSY22} initiated the study of the $k$-core decomposition problem under differential privacy via a connection between low round/depth distributed/parallel graph algorithms and private algorithms with small error bounds. They showed that one can output differentially private approximate $k$-core numbers, while only incurring a multiplicative error of $(2 +η)$ (for any constant $η>0$) and additive error of $\poly(\log(n))/\eps$. In this paper, we revisit this problem. Our main result is an $\eps$-edge differentially private algorithm for $k$-core decomposition which outputs the core numbers with no multiplicative error and $O(\text{log}(n)/\eps)$ additive error. This improves upon previous work by a factor of 2 in the multiplicative error, while giving near-optimal additive error. Our result relies on a novel generalized form of the sparse vector technique, which is especially well-suited for threshold-based graph algorithms; thus, we further strengthen the connection between distributed/parallel graph algorithms and differentially private algorithms.

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arXiv2023

Differentially Private Partial Set Cover with Applications to Facility Location

It was observed in \citet{gupta2009differentially} that the Set Cover problem has strong impossibility results under differential privacy. In our work, we observe that these hardness results dissolve when we turn to the Partial Set Cover problem, where we only need to cover a $ρ$-fraction of the elements in the universe, for some $ρ\in(0,1)$. We show that this relaxation enables us to avoid the impossibility results: under loose conditions on the input set system, we give differentially private algorithms which output an explicit set cover with non-trivial approximation guarantees. In particular, this is the first differentially private algorithm which outputs an explicit set cover. Using our algorithm for Partial Set Cover as a subroutine, we give a differentially private (bicriteria) approximation algorithm for a facility location problem which generalizes $k$-center/$k$-supplier with outliers. Like with the Set Cover problem, no algorithm has been able to give non-trivial guarantees for $k$-center/$k$-supplier-type facility location problems due to the high sensitivity and impossibility results. Our algorithm shows that relaxing the covering requirement to serving only a $ρ$-fraction of the population, for $ρ\in(0,1)$, enables us to circumvent the inherent hardness. Overall, our work is an important step in tackling and understanding impossibility results in private combinatorial optimization.

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arXiv2022

A Simple Differentially Private Algorithm for Global Minimum Cut

In this note, we present a simple differentially private algorithm for the global minimum cut problem using only one call to the exponential mechanism. This problem was first studied by Gupta et al. [2010], and they gave a differentially private algorithm with near-optimal utility guarantees. We improve upon their work in many aspects: our algorithm is simpler, more natural, and more efficient than the one given in Gupta et al. [2010], and furthermore provides slightly better privacy and utility guarantees.

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