Tight Algorithms for Vertex Cover with Hard Capacities on Multigraphs and Hypergraphs

TL;DR

This paper presents an optimal approximation algorithm for the vertex cover problem with hard capacities on hypergraphs, improving previous bounds and employing iterative rounding techniques for efficiency.

Contribution

The paper introduces an f-approximation algorithm for VCHC on hypergraphs, matching the theoretical lower bound and enhancing previous approximation ratios with a faster implementation.

Findings

Achieved an f-approximation ratio for VCHC on hypergraphs.

Improved previous approximation ratios from 2.155 and 2f.

Provided a faster iterative rounding algorithm implementation.

Abstract

In this paper we give a f-approximation algorithm for the minimum unweighted Vertex Cover problem with Hard Capacity constraints (VCHC) on f-hypergraphs. This problem generalizes standard vertex cover for which the best known approximation ratio is also f and cannot be improved assuming the unique game conjecture. Our result is therefore essentially the best possible. This improves over the previous 2.155 (for f=2) and 2f-approximation algorithms by Cheung, Goemans and Wong (CGW). At the heart of our approach is to apply iterative rounding to the problem with ideas coming from several previous works. We also give a faster implementation of the method based on certain iteratively rounding the solution to certain CGW-style covering LPs. We note that independent of this work, Kao [#kao2017iterative] also recently obtained the same result.