On Approximating Four Covering and Packing Problems

TL;DR

This paper investigates the approximability of four related problems—triangle packing, full sibling reconstruction, maximum profit coverage, and 2-coverage—highlighting their differing properties and improving known bounds.

Contribution

It improves inapproximability bounds for triangle packing, answers open questions on sibling reconstruction, and provides tight bounds for maximum profit coverage.

Findings

Enhanced inapproximability bounds for triangle packing.

Resolved open questions on full sibling reconstruction.

Established tight approximation bounds for maximum profit coverage.

Abstract

In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results; this is done by directly transforming the inapproximability gap of Haastad for the problem of maximizing the number of satisfied equations for a set of equations over GF(2) and is interesting in its own right. Our approximability results on the full siblings reconstruction problems answers questions originally posed by Berger-Wolf et al. and our results on the maximum profit coverage problem provides almost matching upper and lower bounds on the approximation ratio, answering a question posed by Hassin and Or.