Approximating Subdense Instances of Covering Problems

TL;DR

This paper investigates the approximability of subdense graph instances for covering problems, introducing new algorithms, PTASs, and hardness results, with some ratios proven optimal under standard assumptions.

Contribution

It presents novel approximation algorithms and PTASs for subdense covering problems, along with the first hardness results, and introduces an improved recursive sampling method for Vertex Cover.

Findings

New approximation algorithms for subdense covering problems.

Existence of PTASs for certain subdense instances.

Proven optimality of some approximation ratios under complexity assumptions.

Abstract

We study approximability of subdense instances of various covering problems on graphs, defined as instances in which the minimum or average degree is Omega(n/psi(n)) for some function psi(n)=omega(1) of the instance size. We design new approximation algorithms as well as new polynomial time approximation schemes (PTASs) for those problems and establish first approximation hardness results for them. Interestingly, in some cases we were able to prove optimality of the underlying approximation ratios, under usual complexity-theoretic assumptions. Our results for the Vertex Cover problem depend on an improved recursive sampling method which could be of independent interest.