O(f) Bi-Approximation for Capacitated Covering with Hard Capacities

TL;DR

This paper introduces an $O(f)$ bi-approximation algorithm for capacitated vertex cover with hard capacities on hypergraphs, balancing the cost and multiplicity augmentation to improve previous bounds.

Contribution

It presents a new $O(f)$ bi-approximation algorithm that offers a trade-off between cost and multiplicity augmentation, improving upon prior $f^2$ bounds.

Findings

Achieves an $O(f)$ bi-approximation for VC-HC.

Provides a trade-off between multiplicity augmentation and cover cost.

Improves previous approximation ratio from $f^2$ to $(1+1/(k-1))(f-1)$.

Abstract

We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph $G=(V,E)$ with a maximum edge size $f$. Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edges can be covered by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an $O(f)$ bi-approximation for VC-HC that gives a trade-off on the number of augmented multiplicity and the cost of the resulting cover. In particular, we show that, by augmenting the available multiplicity by a factor of $k \ge 2$, a~cover with a cost ratio of $\Big(1+\frac{1}{k-1}\Big)(f-1)$ to the optimal cover for the original instance can be obtained. This improves over a previous result, which has a cost ratio of $f^2$ via augmenting the available multiplicity by a factor of $f$.