Searcharxiv⌕ Search

arXiv subjects

Kent Quanrud

Publications and source records attributed to Kent Quanrud.

At least 19 recordsLinked to original sources
arXiv2025

Approximating Directed Connectivity in Almost-Linear Time

We present randomized algorithms that compute $(1+\epsilon)$-approximate minimum global edge and vertex cuts in weighted directed graphs in $O(\log^4(n) / \epsilon)$ and $O(\log^5(n)/\epsilon)$ single-commodity flows, respectively. With the almost-linear time flow algorithm of [CKL+22], this gives almost linear time approximation schemes for edge and vertex connectivity. By setting $\epsilon$ appropriately, this also gives faster exact algorithms for small vertex connectivity. At the heart of these algorithms is a divide-and-conquer technique called "shrink-wrapping" for a certain well-conditioned rooted Steiner connectivity problem. Loosely speaking, for a root $r$ and a set of terminals, shrink-wrapping uses flow to certify the connectivity from a root $r$ to some of the terminals, and for the remaining uncertified terminals, generates an $r$-cut where the sink component both (a) contains the sink component of the minimum $(r,t)$-cut for each uncertified terminal $t$ and (b) has size proportional to the number of uncertified terminals. This yields a divide-and-conquer scheme over the terminals where we can divide the set of terminals and compute their respective minimum $r$-cuts in smaller, contracted subgraphs.

cs.DS↗
arXiv2025

From Hop Reduction to Sparsification for Negative Length Shortest Paths

The textbook algorithm for real-weighted single-source shortest paths takes $O(mn)$ time on a graph with $m$ edges and $n$ vertices. A recent breakthrough algorithm by [Fin24] takes $\tilde{O}(mn^{8/9})$ randomized time. The running time was subsequently improved to $\tilde{O}(mn^{4/5})$ [HJQ25] and then $\tilde{O}(mn^{3/4}+m^{4/5}n)$ [HJQ26]. We build on the algorithms of [Fin24; HJQ25; HJQ26] to obtain faster strongly-polynomial randomized-time algorithms for negative-length shortest paths. An important new technique in this algorithm repurposes previous "hop-reducers" from [Fin24; HJQ26] into "negative edge sparsifiers", reducing the number of negative edges by essentially the same factor by which the "hops" were previously reduced. A simple recursive algorithm based on sparsifying the layered hop reducers of [Fin24] already gives an $\tilde{O}(mn^{\sqrt{3}-1})<O(mn^{.7321})$ randomized running time, improving [HJQ26] uniformly. We also improve the construction of the bootstrapped hop reducers in [HJQ26] by proposing new sparse shortcut graphs replacing the dense shortcut graphs in [HJQ26]. Integrating all three of layered sparsification, recursion, and sparse bootstrapping into the algorithm of [HJQ26] gives new upper bounds of $O(mn^{.7193})$ randomized time for $m\geq n^{1.03456}$ and $O((mn)^{.8620})$ randomized time for $m<n^{1.03456}$. Lastly, concurrent work by [LLRZ25] obtained an $\tilde{O}(n^{2.5})$ randomized time algorithm for the same problem, and along the way improved the running time of the "betweenness reduction" step in Fineman's framework. Dropping in this subroutine as a black box improves the running time of the simple recursive sparsification algorithm to $\tilde{O}(mn^{1/\sqrt{2}})<O(mn^{.70711})$, and a slightly modified recursive sparsification algorithm runs in $O(mn^{.69562})$ randomized time for $m\geq n^{1.0274}$ and $O((mn)^{.85})$ for $m<n^{1.0274}$.

cs.DS↗
arXiv2025

Faster negative length shortest paths by bootstrapping hop reducers

The textbook algorithm for real-weighted single-source shortest paths takes $O(m n)$ time on a graph with $m$ edges and $n$ vertices. The breakthrough algorithm by Fineman [Fin24] takes $\tilde{O}(m n^{8/9})$ randomized time. The running time was subsequently improved to $\tilde{O}(mn^{4/5})$ [HJQ25]. We build on [Fin24; HJQ25] to obtain an $\tilde{O}(m n^{3/4} + m^{4/5} n)$ randomized running time. (Equivalently, $\tilde{O}(mn^{3/4})$ for $m \geq n^{5/4}$, and $\tilde{O}(m^{4/5} n)$ for $m \leq n^{5/4}$.) The main new technique replaces the hop-reducing auxiliary graph from [Fin24] with a bootstrapping process where constant-hop reducers for small subgraphs of the input graph are iteratively amplified and expanded until the desired polynomial-hop reduction is achieved over the entire graph.

cs.DS↗
arXiv2023

Approximate Fully Dynamic Directed Densest Subgraph

We give a fully dynamic algorithm maintaining a $(1-\varepsilon)$-approximate directed densest subgraph in $\tilde{O}(\log^3(n)/\varepsilon^6)$ amortized time or $\tilde{O}(\log^4(n)/\varepsilon^7)$ worst-case time per edge update (where $\tilde{O}$ hides $\log\log$ factors), based on earlier work by Chekuri and Quanrud [arXiv:2210.02611, arXiv:2310.18146]. This result improves on earlier work done by Sawlani and Wang [arXiv:1907.03037], which guarantees $O(\log^5(n)/\varepsilon^7)$ worst case time for edge insertions and deletions.

cs.DS↗
arXiv2023

Adaptive Out-Orientations with Applications

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $α$ of the graph, in, either, an amortised update time of $O(\log^2 n \log α)$, or a worst-case update time of $O(\log^3 n \log α)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely either $O(\log n \log α)$, amortised, or $O(\log ^2 n \log α)$, worst-case time, for the problem of maintaining an edge-orientation with at most $O(α+ \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain an $O(\varepsilon^{-6}\log^3 n \log ρ)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $ρ$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log α)$ worst-case update time algorithm for maintaining a $(1 + \varepsilon) \cdot OPT + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $α$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(α)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $Δ+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(α+\log^2n \log α)$, where $α$ is the current arboricity of the graph.

cs.DS↗
arXiv2023

Independent Sets in Elimination Graphs with a Submodular Objective

Maximum weight independent set (MWIS) admits a $\frac1k$-approximation in inductively $k$-independent graphs and a $\frac{1}{2k}$-approximation in $k$-perfectly orientable graphs. These are a a parameterized class of graphs that generalize $k$-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph $G=(V,E)$ and a non-negative submodular function $f: 2^V \rightarrow \mathbb{R}_+$, the goal is to approximately solve $\max_{S \in \mathcal{I}_G} f(S)$ where $\mathcal{I}_G$ is the set of independent sets of $G$. We obtain an $Ω(\frac1k)$-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least $\frac{1}{e(k+1)}$. This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively $k$-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.

cs.DS↗
arXiv2023

Online Directed Spanners and Steiner Forests

We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $\tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $\tilde{O}(n^{2/3+ε})$-competitive algorithm, and an efficient deterministic $\tilde{O}(k^{1/2+ε})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $\tilde{O}(n^{3/5 + ε})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $\tilde{O}(n^{2/3 + ε})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $\tilde{O}(k^{1/2 + ε})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $\tilde{O}(n^{4/7 + ε})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion.

cs.DS↗
arXiv2023

Convergence to Lexicographically Optimal Base in a (Contra)Polymatroid and Applications to Densest Subgraph and Tree Packing

Boob et al. [1] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Quanrud, and Torres [2] extended the algorithm to general supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper, we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige's quadratic program for finding a lexicographically optimal base in a (contra)polymatroid [3], and a noisy version of the Frank-Wolfe method from convex optimisation [4,5]. This gives us a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [6]. A second contribution of the paper is to understand Thorup's work on ideal tree packing and greedy tree packing [7,8] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige's result and convex optimisation.

cs.DS↗
arXiv2023

Faster exact and approximation algorithms for packing and covering matroids via push-relabel

Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union), computing $k$ disjoint bases (a.k.a. matroid base packing), and covering the elements by $k$ bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement and matroid membership. We obtain improved running times both for general matroids in the independence oracle model and for the graphic matroid. The main thrust of our improvements comes from developing a faster and unifying push-relabel algorithm for the integer-capacitated versions of these problems, building on previous work by Frank and Miklós [FM12]. We then build on this algorithm in two directions. First we develop a faster augmenting path subroutine for $k$-fold matroid union that, when appended to an approximation version of the push-relabel algorithm, gives a faster exact algorithm for some parameters of $k$. In particular we obtain a subquadratic-query running time in the uncapacitated setting for the three basic problems listed above. We also obtain faster approximation algorithms for these problems with real-valued capacities by reducing to small integral capacities via randomized rounding. To this end, we develop a new randomized rounding technique for base covering problems in matroids that may also be of independent interest.

cs.DS↗
arXiv2022

$(1-ε)$-approximate fully dynamic densest subgraph: linear space and faster update time

We consider the problem of maintaining a $(1-ε)$-approximation to the densest subgraph (DSG) in an undirected multigraph as it undergoes edge insertions and deletions (the fully dynamic setting). Sawlani and Wang [SW20] developed a data structure that, for any given $ε> 0$, maintains a $(1-ε)$-approximation with $O(\log^4 n/ε^6)$ worst-case update time for edge operations, and $O(1)$ query time for reporting the density value. Their data structure was the first to achieve near-optimal approximation, and improved previous work that maintained a $(1/4 - ε)$ approximation in amortized polylogarithmic update time [BHNT15]. In this paper we develop a data structure for $(1-ε)$-approximate DSG that improves the one from [SW20] in two aspects. First, the data structure uses linear space improving the space bound in [SW20] by a logarithmic factor. Second, the data structure maintains a $(1-ε)$-approximation in amortized $O(\log^2 n/ε^4)$ time per update while simultaneously guaranteeing that the worst case update time is $O(\log^3 n \log \log n/ε^6)$. We believe that the space and update time improvements are valuable for current large scale graph data sets. The data structure extends in a natural fashion to hypergraphs and yields improvements in space and update times over recent work [BBCG22] that builds upon [SW20].

cs.DS↗
arXiv2021

Minimum Cuts in Directed Graphs via Partial Sparsification

We give an algorithm to find a minimum cut in an edge-weighted directed graph with $n$ vertices and $m$ edges in $\tilde O(n\cdot \max(m^{2/3}, n))$ time. This improves on the 30 year old bound of $\tilde O(nm)$ obtained by Hao and Orlin for this problem. Our main technique is to reduce the directed mincut problem to $\tilde O(\min(n/m^{1/3}, \sqrt{n}))$ calls of {\em any} maxflow subroutine. Using state-of-the-art maxflow algorithms, this yields the above running time. Our techniques also yield fast {\em approximation} algorithms for finding minimum cuts in directed graphs. For both edge and vertex weighted graphs, we give $(1+ε)$-approximation algorithms that run in $\tilde O(n^2 / ε^2)$ time.

cs.DS↗
arXiv2021

Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a $(1+ε)$ approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate $B$ such that $B \le B^* \le \lceil (1+ε)B \rceil +1$ where $B^*$ is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.

cs.DS↗
arXiv2021

Faster Algorithms for Rooted Connectivity in Directed Graphs

We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time $\tilde{O}(n^2)$. For rooted edge connectivity this is the first algorithm to improve on the $Ω(n^3)$ time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems. We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a $(1 + ε)$-approximation for rooted vertex connectivity in $\tilde{O}(nW/ε)$ time where $W$ is the total vertex weight (assuming integral vertex weights); in particular this yields an $\tilde{O}(n^2/ε)$ time randomized algorithm for unweighted graphs. This translates to a $\tilde{O}(κnW)$ time exact algorithm where $κ$ is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity. Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [9] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [24] and Forster et al. [7] for vertex connectivity.

cs.DS↗
arXiv2021

Fast Approximations for Rooted Connectivity in Weighted Directed Graphs

We consider approximations for computing minimum weighted cuts in directed graphs. We consider both rooted and global minimum cuts, and both edge-cuts and vertex-cuts. For these problems we give randomized Monte Carlo algorithms that compute a $(1+ε)$-approximate minimum cut in $\tilde{O}(n^2 / ε^2)$ time. These results extend and build on recent work [4] that obtained exact algorithms with similar running times in directed graphs with small integer capacities.

cs.DS↗
arXiv2021

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

Li and Panigrahi, in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time $o(mn)$. They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of $s$-$t$ minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions.

cs.DS↗
arXiv2019

Computing Circle Packing Representations of Planar Graphs

The Circle Packing Theorem states that every planar graph can be represented as the tangency graph of a family of internally-disjoint circles. A well-known generalization is the Primal-Dual Circle Packing Theorem for 3-connected planar graphs. The existence of these representations has widespread applications in theoretical computer science and mathematics; however, the algorithmic aspect has received relatively little attention. In this work, we present an algorithm based on convex optimization for computing a primal-dual circle packing representation of maximal planar graphs, i.e. triangulations. This in turn gives an algorithm for computing a circle packing representation of any planar graph. Both take $\widetilde{O}(n \log(R/\varepsilon))$ expected run-time to produce a solution that is $\varepsilon$ close to a true representation, where $R$ is the ratio between the maximum and minimum circle radius in the true representation.

cs.CG↗
arXiv2019

Adversarial Delays in Online Strongly-Convex Optimization

We consider the problem of strongly-convex online optimization in presence of adversarial delays; in a T-iteration online game, the feedback of the player's query at time t is arbitrarily delayed by an adversary for d_t rounds and delivered before the game ends, at iteration t+d_t-1. Specifically for \algo{online-gradient-descent} algorithm we show it has a simple regret bound of \Oh{\sum_{t=1}^T \log (1+ \frac{d_t}{t})}. This gives a clear and simple bound without resorting any distributional and limiting assumptions on the delays. We further show how this result encompasses and generalizes several of the existing known results in the literature. Specifically it matches the celebrated logarithmic regret \Oh{\log T} when there are no delays (i.e. d_t = 1) and regret bound of \Oh{τ\log T} for constant delays d_t = τ.

cs.LG↗