Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

TL;DR

This paper introduces the first deterministic data structures for fully dynamic approximate vertex cover and matching problems, achieving sublinear update times and settling an open problem in the field.

Contribution

It presents deterministic algorithms for maintaining approximate solutions to vertex cover and matching with improved update times, including near-optimal bounds for vertex cover.

Findings

Deterministic data structures for approximate vertex cover and matching.

Achieves sublinear amortized update times, including near-optimal bounds.

Provides practical algorithms with theoretical guarantees.

Abstract

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m}\,)$ time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a $(2+\eps)$ approximation in $O(\log n/\eps^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\eps)$ approximation in $O(\min(\sqrt{n}/\epsilon, m^{1/3}/\eps^2))$ {\em amortized} time per update, and a $(4+\eps)$ approximation in $O(m^{1/3}/\eps^2)$ {\em worst-case} time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld from STOC' 2010.