arXiv subjects

Shay Solomon

Publications and source records attributed to Shay Solomon.

At least 37 records · Page 2Linked to original sources

arXiv2024

Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985]. In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years.

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arXiv2024

Optimal Euclidean Tree Covers

A $(1+\varepsilon)\textit{-stretch tree cover}$ of a metric space is a collection of trees, where every pair of points has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated $\textit{Dumbbell Theorem}$ [Arya et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d} \cdot \log(1/\varepsilon))$ trees, where the $O_d$ notation suppresses terms that depend solely on the dimension~$d$. The running time of their construction is $O_d(n \log n \cdot \frac{\log(1/\varepsilon)}{\varepsilon^{d}} + n \cdot \varepsilon^{-2d})$. Since the same point may occur in multiple levels of the tree, the $\textit{maximum degree}$ of a point in the tree cover may be as large as $Ω(\log Φ)$, where $Φ$ is the aspect ratio of the input point set. In this work we present a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d+1} \cdot \log(1/\varepsilon))$ trees, which is optimal (up to the $\log(1/\varepsilon)$ factor). Moreover, the maximum degree of points in any tree is an $\textit{absolute constant}$ for any $d$. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a $(1+\varepsilon)$-stretch $\textit{Steiner}$ tree cover (that may use Steiner points) with $O_d(\varepsilon^{(-d+1)/{2}} \cdot \log(1/\varepsilon))$ trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive $O_d(n \log n)$ term; this improves over the running time underlying the Dumbbell Theorem.

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arXiv2024

Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the $Ω(\log n)$ Lightness Barrier

An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+ε)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+ε)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"ıvely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $Ω(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $Ω(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $ε> 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+ε)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$.

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arXiv2024

Arboricity-Dependent Algorithms for Edge Coloring

The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a $(Δ+ O(Δ^{1 - μ}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $μ> 0$, where $Δ$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(Δ+ O(α))$-edge coloring in $\tilde O(1)$ amortized update time, where $α$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.

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arXiv2023

Dynamic $((1+ε)\ln n)$-Approximation Algorithms for Minimum Set Cover and Dominating Set

The minimum set cover (MSC) problem admits two classic algorithms: a greedy $\ln n$-approximation and a primal-dual $f$-approximation, where $n$ is the universe size and $f$ is the maximum frequency of an element. Both algorithms are simple and efficient, and remarkably -- one cannot improve these approximations under hardness results by more than a factor of $(1+ε)$, for any constant $ε> 0$. In their pioneering work, Gupta et al. [STOC'17] showed that the greedy algorithm can be dynamized to achieve $O(\log n)$-approximation with update time $O(f \log n)$. Building on this result, Hjuler et al. [STACS'18] dynamized the greedy minimum dominating set (MDS) algorithm, achieving a similar approximation with update time $O(Δ\log n)$ (the analog of $O(f \log n)$), albeit for unweighted instances. The approximations of both algorithms, which are the state-of-the-art, exceed the static $\ln n$-approximation by a rather large constant factor. In sharp contrast, the current best dynamic primal-dual MSC algorithms achieve fast update times together with an approximation that exceeds the static $f$-approximation by a factor of (at most) $1+ε$, for any $ε> 0$. This paper aims to bridge the gap between the best approximation factor of the dynamic greedy MSC and MDS algorithms and the static $\ln n$ bound. We present dynamic algorithms for weighted greedy MSC and MDS with approximation $(1+ε)\ln n$ for any $ε> 0$, while achieving the same update time (ignoring dependencies on $ε$) of the best previous algorithms (with approximation significantly larger than $\ln n$). Moreover, [...]

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arXiv2023

Covering Planar Metrics (and Beyond): O(1) Trees Suffice

While research on the geometry of planar graphs has been active in the past decades, many properties of planar metrics remain mysterious. This paper studies a fundamental aspect of the planar graph geometry: covering planar metrics by a small collection of simpler metrics. Specifically, a \emph{tree cover} of a metric space $(X, δ)$ is a collection of trees, so that every pair of points $u$ and $v$ in $X$ has a low-distortion path in at least one of the trees. The celebrated "Dumbbell Theorem" [ADMSS95] states that any low-dimensional Euclidean space admits a tree cover with $O(1)$ trees and distortion $1+\varepsilon$, for any fixed $\varepsilon \in (0,1)$. This result has found numerous algorithmic applications, and has been generalized to the wider family of doubling metrics [BFN19]. Does the same result hold for planar metrics? A positive answer would add another evidence to the well-observed connection between Euclidean/doubling metrics and planar metrics. In this work, we answer this fundamental question affirmatively. Specifically, we show that for any given fixed $\varepsilon \in (0,1)$, any planar metric can be covered by $O(1)$ trees with distortion $1+\varepsilon$. Our result for planar metrics follows from a rather general framework: First we reduce the problem to constructing tree covers with \emph{additive distortion}. Then we introduce the notion of \emph{shortcut partition}, and draw connection between shortcut partition and additive tree cover. Finally we prove the existence of shortcut partition for any planar metric, using new insights regarding the grid-like structure of planar graphs. [...]

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arXiv2023

Nibbling at Long Cycles: Dynamic (and Static) Edge Coloring in Optimal Time

We consider the problem of maintaining a $(1+ε)Δ$-edge coloring in a dynamic graph $G$ with $n$ nodes and maximum degree at most $Δ$. The state-of-the-art update time is $O_ε(\text{polylog}(n))$, by Duan, He and Zhang [SODA'19] and by Christiansen [STOC'23], and more precisely $O(\log^7 n/ε^2)$, where $Δ= Ω(\log^2 n / ε^2)$. The following natural question arises: What is the best possible update time of an algorithm for this task? More specifically, \textbf{ can we bring it all the way down to some constant} (for constant $ε$)? This question coincides with the \emph{static} time barrier for the problem: Even for $(2Δ-1)$-coloring, there is only a naive $O(m \log Δ)$-time algorithm. We answer this fundamental question in the affirmative, by presenting a dynamic $(1+ε)Δ$-edge coloring algorithm with $O(\log^4 (1/ε)/ε^9)$ update time, provided $Δ= Ω_ε(\text{polylog}(n))$. As a corollary, we also get the first linear time (for constant $ε$) \emph{static} algorithm for $(1+ε)Δ$-edge coloring; in particular, we achieve a running time of $O(m \log (1/ε)/ε^2)$. We obtain our results by carefully combining a variant of the \textsc{Nibble} algorithm from Bhattacharya, Grandoni and Wajc [SODA'21] with the subsampling technique of Kulkarni, Liu, Sah, Sawhney and Tarnawski [STOC'22].

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arXiv2023

Resolving the Steiner Point Removal Problem in Planar Graphs via Shortcut Partitions

Recently the authors [CCLMST23] introduced the notion of shortcut partition of planar graphs and obtained several results from the partition, including a tree cover with $O(1)$ trees for planar metrics and an additive embedding into small treewidth graphs. In this note, we apply the same partition to resolve the Steiner point removal (SPR) problem in planar graphs: Given any set $K$ of terminals in an arbitrary edge-weighted planar graph $G$, we construct a minor $M$ of $G$ whose vertex set is $K$, which preserves the shortest-path distances between all pairs of terminals in $G$ up to a constant factor. This resolves in the affirmative an open problem that has been asked repeatedly in literature.

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arXiv2023

Shortcut Partitions in Minor-Free Graphs: Steiner Point Removal, Distance Oracles, Tree Covers, and More

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenković, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices $u$ and $v$ in the graph, there exists a path between $u$ and $v$ that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch $1+\varepsilon$ and $O(1)$ many trees for any fixed $\varepsilon \in (0,1)$. However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs. In this work, we breach the "planarity barrier" to construct a shortcut partition for $K_r$-minor-free graphs for any $r$. To this end, we take a completely different approach -- our key contribution is a novel deterministic variant of the cop decomposition in minor-free graphs [And86, AGG14]. Our shortcut partition for $K_r$-minor-free graphs yields several direct applications. Most notably, we construct the first optimal distance oracle for $K_r$-minor-free graphs, with $1+\varepsilon$ stretch, linear space, and constant query time for any fixed $\varepsilon \in (0,1)$. The previous best distance oracle [AG06] uses $O(n\log n)$ space and $O(\log n)$ query time, and its construction relies on Robertson-Seymour structural theorem and other sophisticated tools. We also obtain the first tree cover of $O(1)$ size for minor-free graphs with stretch $1+\varepsilon$, while the previous best $(1+\varepsilon)$-tree cover has size $O(\log^2 n)$ [BFN19].

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arXiv2023

A Unified Framework for Light Spanners

Seminal works on light spanners over the years have provided spanners with optimal lightness in various graph classes, such as general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (i) The runtimes of these constructions are almost always sub-optimal and usually far from optimal. (ii) These constructions are optimal in the standard and crude sense but not in a refined sense that takes into account a wider range of involved parameters. (iii) The techniques are ad hoc per graph class and thus can't be applied broadly. This work aims at addressing these shortcomings by presenting a unified framework of light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. First, we apply our framework to design fast constructions with optimal lightness for several graph classes. Second, we apply our framework to achieve more refined optimality bounds for several graph classes, i.e., the bounds remain optimal when taking into account a wider range of involved parameters, most notably $ε$. Our new constructions are significantly better than the state-of-the-art for every examined graph class.

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arXiv2022

Can't See The Forest for the Trees: Navigating Metric Spaces by Bounded Hop-Diameter Spanners

Spanners for metric spaces have been extensively studied, both in general metrics and in restricted classes, perhaps most notably in low-dimensional Euclidean spaces -- due to their numerous applications. Euclidean spanners can be viewed as means of compressing the $\binom{n}{2}$ pairwise distances of a $d$-dimensional Euclidean space into $O(n) = O_{ε,d}(n)$ spanner edges, so that the spanner distances preserve the original distances to within a factor of $1+ε$, for any $ε> 0$. Moreover, one can compute such spanners in optimal $O(n \log n)$ time. Once the spanner has been computed, it serves as a "proxy" overlay network, on which the computation can proceed, which gives rise to huge savings in space and other important quality measures. On the negative side, by working on the spanner rather than the original metric, one loses the key property of being able to efficiently "navigate" between pairs of points. While in the original metric, one can go from any point to any other via a direct edge, it is unclear how to efficiently navigate in the spanner: How can we translate the existence of a "good" path into an efficient algorithm finding it? Moreover, usually by "good" path we mean a path whose weight approximates the original distance between its endpoints -- but a priori the number of edges (or "hops") in the path could be huge. To control the hop-length of paths, one can try to upper bound the spanner's hop-diameter, but naturally bounded hop-diameter spanners are more complex than spanners with unbounded hop-diameter, which might render the algorithmic task of efficiently finding good paths more challenging. The original metric enables us to navigate optimally -- a single hop (for any two points) with the exact distance, but the price is high -- $Θ(n^2)$ edges. [...]

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arXiv2021

Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

In STOC'95 [ADMSS'95] Arya et al. showed that any set of $n$ points in $\mathbb R^d$ admits a $(1+ε)$-spanner with hop-diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges). They also gave a general upper bound tradeoff of hop-diameter at most $k$ and $O(n α_k(n))$ edges, for any $k \ge 2$. The function $α_k$ is the inverse of a certain Ackermann-style function at the $\lfloor k/2 \rfloor$th level of the primitive recursive hierarchy, where $α_0(n) = \lceil n/2 \rceil$, $α_1(n) = \left\lceil \sqrt{n} \right\rceil$, $α_2(n) = \lceil \log{n} \rceil$, $α_3(n) = \lceil \log\log{n} \rceil$, $α_4(n) = \log^* n$, $α_5(n) = \lfloor \frac{1}{2} \log^*n \rfloor$, \ldots. Roughly speaking, for $k \ge 2$ the function $α_{k}$ is close to $\lfloor \frac{k-2}{2} \rfloor$-iterated log-star function, i.e., $\log$ with $\lfloor \frac{k-2}{2} \rfloor$ stars. Also, $α_{2α(n)+4}(n) \le 4$, where $α(n)$ is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases $k = 2$ and $k = 3$. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of $k$. In this paper we prove a tight lower bound for any constant $k$: For any fixed $ε> 0$, any $(1+ε)$-spanner for the uniform line metric with hop-diameter at most $k$ must have at least $Ω(n α_k(n))$ edges.

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arXiv2021

A Unified Framework of Light Spanners II: Fine-Grained Optimality

Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly. (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (3) These constructions are optimal in the standard and crude sense, but not in a refined sense that takes into account a wider range of involved parameters. This work aims at addressing these shortcomings by presenting a unified framework of light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. Our framework is developed in two papers. The current paper is the second of the two -- it builds on the basis of the unified framework laid in the first paper, and then strengthens it to achieve more refined optimality bounds for several graph classes. Among various applications and implications of our framework, we highlight here the following: For $K_r$-minor-free graphs, we provide a $(1+ε)$-spanner with lightness $\tilde{O}_{r,ε}( \frac{r}ε + \frac{1}{ε^2})$, improving the lightness bound $\tilde{O}_{r,ε}( \frac{r}{ε^3})$ of Borradaile, Le and Wulff-Nilsen. We complement our upper bound with a lower bound construction, for which any $(1+ε)$-spanner must have lightness $Ω(\frac{r}ε + \frac{1}{ε^2})$. We note that the quadratic dependency on $1/ε$ we proved here is surprising, as the prior work suggested that the dependency on $ε$ should be $1/ε$.

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arXiv2021

Near-Optimal Distributed Implementations of Dynamic Algorithms for Symmetry-Breaking Problems

The field of dynamic graph algorithms aims at achieving a thorough understanding of real-world networks whose topology evolves with time. Traditionally, the focus has been on the classic sequential, centralized setting where the main quality measure of an algorithm is its update time, i.e. the time needed to restore the solution after each update. While real-life networks are very often distributed across multiple machines, the fundamental question of finding efficient dynamic, distributed graph algorithms received little attention to date. The goal in this setting is to optimize both the round and message complexities incurred per update step, ideally achieving a message complexity that matches the centralized update time in $O(1)$ (perhaps amortized) rounds. Toward initiating a systematic study of dynamic, distributed algorithms, we study some of the most central symmetry-breaking problems: maximal independent set (MIS), maximal matching/(approx-) maximum cardinality matching (MM/MCM), and $(Δ+ 1)$-vertex coloring. This paper focuses on dynamic, distributed algorithms that are deterministic, and in particular -- robust against an adaptive adversary. Most of our focus is on our MIS algorithm, which achieves $O\left(m^{2/3}\log^2 n\right)$ amortized messages in $O\left(\log^2 n\right)$ amortized rounds in the Congest model. Notably, the amortized message complexity of our algorithm matches the amortized update time of the best-known deterministic centralized MIS algorithm by Gupta and Khan [SOSA'21] up to a polylog $n$ factor. The previous best deterministic distributed MIS algorithm, by Assadi et al. [STOC'18], uses $O(m^{3/4})$ amortized messages in $O(1)$ amortized rounds, i.e., we achieve a polynomial improvement in the message complexity by a polylog $n$ increase to the round complexity; moreover, the algorithm of Assadi et al. makes an implicit assumption that the [...]

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arXiv2021

Maintaining an EDCS in General Graphs: Simpler, Density-Sensitive and with Worst-Case Time Bounds

In their breakthrough ICALP'15 paper, Bernstein and Stein presented an algorithm for maintaining a $(3/2+ε)$-approximate maximum matching in fully dynamic {\em bipartite} graphs with a {\em worst-case} update time of $O_ε(m^{1/4})$; we use the $O_ε$ notation to suppress the $ε$-dependence. Their main technical contribution was in presenting a new type of bounded-degree subgraph, which they named an {\em edge degree constrained subgraph (EDCS)}, which contains a large matching -- of size that is smaller than the maximum matching size of the entire graph by at most a factor of $3/2+ε$. They demonstrate that the EDCS can be maintained with a worst-case update time of $O_ε(m^{1/4})$, and their main result follows as a direct corollary. In their followup SODA'16 paper, Bernstein and Stein generalized their result for general graphs, achieving the same update time of $O_ε(m^{1/4})$, albeit with an amortized rather than worst-case bound. To date, the best {\em deterministic} worst-case update time bound for {\em any} better-than-2 approximate matching is $O(\sqrt{m})$ [Neiman and Solomon, STOC'13], [Gupta and Peng, FOCS'13]; allowing randomization (against an oblivious adversary) one can achieve a much better (still polynomial) update time for approximation slightly below 2 [Behnezhad, Lacki and Mirrokni, SODA'20]. In this work we\footnote{\em quasi nanos, gigantium humeris insidentes} simplify the approach of Bernstein and Stein for bipartite graphs, which allows us to generalize it for general graphs while maintaining the same bound of $O_ε(m^{1/4})$ on the {\em worst-case} update time. Moreover, our approach is {\em density-sensitive}: If the {\em arboricity} of the dynamic graph is bounded by $α$ at all times, then the worst-case update time of the algorithm is $O_ε(\sqrtα)$.

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arXiv2021

Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial

We revisit the minimum dominating set problem on graphs with arboricity bounded by $α$. Bansal and Umboh [BU17] gave an $O(α)$-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [KY14, You14, AZO19, Qua10]. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [LW10], and Jones et al. [JLR+13], achieve an approximation factor of $O(α^2)$ in linear time. There is a similar situation in the distributed setting: While there is an $O(\log^2 n)$-round LP-based $O(α)$-approximation algorithm implied in [KMW06], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(α^2)$-approximation within $O(\log n)$ rounds. We address the questions of whether one can achieve an $O(α)$-approximation algorithm that is not LP-based, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. More specifically, our contribution is two-fold: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an $O(α)$-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(α)$-approximation algorithm in the CONGEST model that runs in $O(α\log n )$ rounds with high probability.

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arXiv2021

Near-Optimal Spanners for General Graphs in (Nearly) Linear Time

Let $G = (V,E,w)$ be a weighted undirected graph on $|V| = n$ vertices and $|E| = m$ edges, let $k \ge 1$ be any integer, and let $ε< 1$ be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdős' girth conjecture and disregarding the $ε$-dependencies.) - There are (deterministic) algorithms for constructing $(2k-1)(1+ε)$-spanners for $G$ with a near-optimal sparsity of $O(n^{1/k} \log(1/ε)/ε))$. The first algorithm can be implemented in the pointer-machine model within time $O(mα(m,n) \log(1/ε)/ε) + SORT(m))$, where $α( , )$ is the two-parameter inverse-Ackermann function and $SORT(m)$ is the time needed to sort $m$ integers. The second algorithm can be implemented in the WORD RAM model within time $O(m \log(1/ε)/ε))$. - There is a (deterministic) algorithm for constructing a $(2k-1)(1+ε)$-spanner for $G$ that achieves a near-optimal bound of $O(n^{1/k}\mathrm{poly}(1/ε))$ on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time $O(mα(m,n) \mathrm{poly}(1/ε) + SORT(m))$ and in the WORD RAM model within time $O(m α(m,n) \mathrm{poly}(1/ε))$. The previous fastest constructions of $(2k-1)(1+ε)$-spanners with near-optimal sparsity incur a runtime of is $O(\min\{m(n^{1+1/k}) + n\log n,k n^{2+1/k}\})$, even regardless of the lightness. Importantly, the greedy spanner for stretch $2k-1$ has sparsity $O(n^{1/k})$ -- with no $ε$-dependence whatsoever, but its runtime is $O(m(n^{1+1/k} + n\log n))$. Moreover, the state-of-the-art lightness bound of any $(2k-1)$-spanner is poor, even regardless of the sparsity and runtime.

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arXiv2021

Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach

In the (fully) dynamic set cover problem, we have a collection of $m$ sets from a universe of size $n$ that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an $O(f^2)$ update time algorithm for this problem that achieves an $f$-approximation, where $f$ is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed $f$. This is the first algorithm for dynamic set cover with approximation ratio that {exactly} matches $f$ (as opposed to {almost} $f$ in prior work), as well as the first one with runtime \emph{independent of $n,m$} (for any approximation factor of $o(f^3)$). Prior to our work, the state-of-the-art algorithms for this problem were $O(f^2)$ update time algorithms of Gupta et al. [STOC'17] and Bhattacharya et al. [IPCO'17] with $O(f^3)$ approximation, and the recent algorithm of Bhattacharya et al. [FOCS'19] with $O(f \cdot \log{n}/ε^2)$ update time and $(1+ε) \cdot f$ approximation, improving the $O(f^2 \cdot \log{n}/ε^5)$ bound of Abboud et al. [STOC'19]. The key technical ingredient of our work is an algorithm for maintaining a {maximal} matching in a dynamic hypergraph of rank $r$, where each hyperedge has at most $r$ vertices, which undergoes hyperedge insertions and deletions in $O(r^2)$ amortized update time; our algorithm is randomized, and the bound on the update time holds in expectation and with high probability. This result generalizes the maximal matching algorithm of Solomon [FOCS'16] with constant update time in ordinary graphs to hypergraphs, and is of independent merit; the previous state-of-the-art algorithms for set cover do not translate to (integral) matchings for hypergraphs, let alone a maximal one. Our quantitative result for the set cover problem is [...]

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