Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

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 $(2+\epsilon)$ approximation in $O(\log n/\epsilon^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\epsilon)$ approximation in $O(\min(\sqrt{n}/\epsilon, m^{1/3}/\epsilon^2)$ amortized time per update and a $(4+\epsilon)$ approximation in $O(m^{1/3}/\epsilon^2)$ 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 [in 42nd ACM Symposium on Theory of Computing, Cambridge, MA, ACM, 2010, pp. 457--464].