A Massively Parallel Algorithm for Minimum Weight Vertex Cover

TL;DR

This paper introduces a massively parallel algorithm that efficiently computes a near-optimal weighted vertex cover in logarithmic rounds, addressing a key gap in existing algorithms for weighted problems.

Contribution

It presents the first near-linear memory MPC algorithm achieving a $(2+ ext{epsilon})$-approximation for minimum-weight vertex cover in $O( ext{log log d})$ rounds, extending prior work to the weighted case.

Findings

Achieves $(2+\varepsilon)$-approximation in $O(\log\log d)$ rounds.

Fills the gap for weighted vertex cover in MPC model.

Improves upon previous $O(\log n)$ algorithms for weighted cases.

Abstract

We present a massively parallel algorithm, with near-linear memory per machine, that computes a $(2+\varepsilon)$-approximation of minimum-weight vertex cover in $O(\log\log d)$ rounds, where $d$ is the average degree of the input graph. Our result fills the key remaining gap in the state-of-the-art MPC algorithms for vertex cover and matching problems; two classic optimization problems, which are duals of each other. Concretely, a recent line of work---by Czumaj et al. [STOC'18], Ghaffari et al. [PODC'18], Assadi et al. [SODA'19], and Gamlath et al. [PODC'19]---provides $O(\log\log n)$ time algorithms for $(1+\varepsilon)$-approximate maximum weight matching as well as for $(2+\varepsilon)$-approximate minimum cardinality vertex cover. However, the latter algorithm does not work for the general weighted case of vertex cover, for which the best known algorithm remained at $O(\log n)$ time complexity.