arXiv subjects

Amitai Uzrad

Publications and source records attributed to Amitai Uzrad.

5 recordsLinked to original sources

arXiv2026

Engineering Algorithms for Dynamic Greedy Set Cover

In the dynamic set cover problem, the input is a dynamic universe of elements and a fixed collection of sets. As elements are inserted or deleted, the goal is to efficiently maintain an approximate minimum set cover. While the past decade has seen significant theoretical breakthroughs for this problem, a notable gap remains between theoretical design and practical performance, as no comprehensive experimental study currently exists to validate these results. In this paper, we bridge this gap by implementing and evaluating four greedy-based dynamic algorithms across a diverse range of real-world instances. We derive our implementations from state-of-the-art frameworks (such as GKKP, STOC 2017; SU, STOC 2023; SUZ, FOCS 2024), which we simplify by identifying and modifying intricate subroutines that optimize asymptotic bounds but hinder practical performance. We evaluate these algorithms based on solution quality (set cover size) and efficiency, which comprises update time (the time required to update the solution following each insertion or deletion) and recourse (the number of changes made to the solution per update). Each algorithm uses a parameter $β$ to balance quality against efficiency; we investigate the influence of this tradeoff parameter on each algorithm and then perform a comparative analysis to evaluate the algorithms against each other. Our results provide the first practical insights into which algorithmic strategies provide the most value in realistic scenarios.

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arXiv2025

Dynamic Set Cover with Worst-Case Recourse

In the dynamic set cover (SC) problem, the input is a dynamic universe of at most $n$ elements and a fixed collection of $m$ sets, where each element belongs to at most $f$ sets and each set has cost in $[1/C, 1]$. The objective is to efficiently maintain an approximate minimum SC under element updates; efficiency is primarily measured by the update time, but another important parameter is the recourse (number of changes to the solution per update). Ideally, one would like to achieve low worst-case bounds on both update time and recourse. One can achieve approximation $(1+ε)\ln n$ (greedy-based) or $(1+ε)f$ (primal-dual-based) with worst-case update time $O(f\log n)$ (ignoring $ε$ dependencies). However, despite a large body of work, no algorithm with low update time (even amortized) and nontrivial worst-case recourse is known, even for unweighted instances ($C = 1$)! We remedy this by providing a transformation that, given as a black-box a SC algorithm with approximation $α$ and update time $T$, returns a set cover algorithm with approximation $(2 + ε)α$, update time $O(T + αC)$, and worst-case recourse $O(αC)$. Our main results are obtained by leveraging this transformation for constant $C$:...

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arXiv2024

A Lossless Deamortization for Dynamic Greedy Set Cover

The dynamic set cover problem has been subject to growing research attention in recent years. In this problem, we are given as input a dynamic universe of at most $n$ elements and a fixed collection of $m$ sets, where each element appears in a most $f$ sets and the cost of each set is in $[1/C, 1]$, and the goal is to efficiently maintain an approximate minimum set cover under element updates. Two algorithms that dynamize the classic greedy algorithm are known, providing $O(\log n)$ and $((1+ε)\ln n)$-approximation with amortized update times $O(f \log n)$ and $O(\frac{f \log n}{ε^5})$, respectively [GKKP (STOC'17); SU (STOC'23)]. The question of whether one can get approximation $O(\log n)$ (or even worse) with low worst-case update time has remained open -- only the naive $O(f \cdot n)$ time bound is known, even for unweighted instances. In this work we devise the first amortized greedy algorithm that is amenable to an efficient deamortization, and also develop a lossless deamortization approach suitable for the set cover problem, the combination of which yields a $((1+ε)\ln n)$-approximation algorithm with a worst-case update time of $O(\frac{f\log n}{ε^2})$. Our worst-case time bound -- the first to break the naive $O(f \cdot n)$ bound -- matches the previous best amortized bound, and actually improves its $ε$-dependence. Further, to demonstrate the applicability of our deamortization approach, we employ it, in conjunction with the primal-dual amortized algorithm of [BHN (FOCS'19)], to obtain a $((1+ε)f)$-approximation algorithm with a worst-case update time of $O(\frac{f\log n}{ε^2})$, improving over the previous best bound of $O(\frac{f \cdot \log^2(Cn)}{ε^3})$ [BHNW (SODA'21)]. Finally, as direct implications of our results for set cover, we [...]

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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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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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