Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2009.04556

Sensitivity Analysis of the Maximum Matching Problem

Yuichi Yoshida·Samson Zhou
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the sensitivity of algorithms for the maximum matching problem against edge and vertex modifications. Algorithms with low sensitivity are desirable because they are robust to edge failure or attack. In this work, we show a randomized $(1-ε)$-approximation algorithm with worst-case sensitivity $O_ε(1)$, which substantially improves upon the $(1-ε)$-approximation algorithm of Varma and Yoshida (arXiv 2020) that obtains average sensitivity $n^{O(1/(1+ε^2))}$ sensitivity algorithm, and show a deterministic $1/2$-approximation algorithm with sensitivity $\exp(O(\log^*n))$ for bounded-degree graphs. We show that any deterministic constant-factor approximation algorithm must have sensitivity $Ω(\log^* n)$. Our results imply that randomized algorithms are strictly more powerful than deterministic ones in that the former can achieve sensitivity independent of $n$ whereas the latter cannot. We also show analogous results for vertex sensitivity, where we remove a vertex instead of an edge. As an application of our results, we give an algorithm for the online maximum matching with $O_ε(n)$ total replacements in the vertex-arrival model. By comparison, Bernstein et al. (J. ACM 2019) gave an online algorithm that always outputs the maximum matching, but only for bipartite graphs and with $O(n\log n)$ total replacements. Finally, we introduce the notion of normalized weighted sensitivity, a natural generalization of sensitivity that accounts for the weights of deleted edges. We show that if all edges in a graph have polynomially bounded weight, then given a trade-off parameter $α>2$, there exists an algorithm that outputs a $\frac{1}{4α}$-approximation to the maximum weighted matching in $O(m\log_α n)$ time, with normalized weighted sensitivity $O(1)$. See paper for full abstract.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuichi Yoshida, Samson Zhou. 2020-09-09. Sensitivity Analysis of the Maximum Matching Problem. https://arxiv.org/abs/2009.04556

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Computational complexity of the recoverable robust shortest path problem in acyclic digraphs under interval budgeted uncertainty

In this paper, we consider the recoverable robust shortest path problem in acyclic digraphs, employing interval budgeted uncertainty to model uncertain second-stage costs. For the continuous budgeted uncertainty model, we prove that the problem is strongly NP-hard even in layered digraphs. Furthermore, we show that in general acyclic digraphs, the problem cannot be approximated within any constant factor unless $\mathrm{P} = \mathrm{NP}$, nor can it be approximated within a factor of $2^{\log^{1-ε} n}$ for any $ε> 0$ unless $\mathrm{NP} \subseteq \mathrm{DTIME}(n^{\mathrm{poly} \log n})$. For the discrete budgeted uncertainty model, we show that the problem is not approximable unless $\mathrm{P} = \mathrm{NP}$, even in layered digraphs. Finally, we establish that under continuous budgeted uncertainty, the integrality gap of a relaxation allowing a fractional first-stage solution is at least $Ω(\sqrt{n})$.

cs.DS↗
arXiv2026

Odd and Even Harder Problems on Cycle-Factors

For a graph (undirected or directed), a cycle-factor is a collection of vertex-disjoint cycles covering the entire vertex set. Cycle-factors subject to parity constraints arise naturally in the study of structural graph theory and algorithmic complexity. In this work, we study four variants of the problem of finding a cycle-factor subject to the following parity constraints: (1) all cycles are odd, (2) all cycles are even, (3) at least one cycle is odd, and (4) at least one cycle is even. We show that all the variants of the problem are NP-complete both in undirected and directed graphs, even if each vertex is incident to at most three edges. We also prove that the first two variants are NP-complete even for planar directed graphs.

cs.DS↗
arXiv2026

An $n^{8/5+o(1)}$-Time $Ω(λ^3)$-Approximation for Longest Common Subsequence

Let $λ$ denote the ratio of the length of a longest common subsequence of two length-$n$ strings to $n$. Rubinstein, Seddighin, Song and Sun [RSSS19] gave an $Ω(λ^3)$-approximation for LCS running in $\widetilde O(n^{39/20})$ time, where $39/20=1.95$. Song [Son19] mentioned that improving the $n^{1.95}$ running time is an interesting open question. We give an algorithm that computes an $Ω(λ^3)$-approximation of the longest common subsequence in $n^{8/5+o(1)}$ time. This improves the exponent $1.95$ to $1.6+o(1)$.

cs.DS↗