Approximation algorithms for hitting subgraphs

TL;DR

This paper investigates when the straightforward $k$-factor approximation for the $H$-hitting set problem can be improved for specific fixed graphs $H$, aiming to find better algorithms for certain subgraph hitting problems.

Contribution

The paper explores conditions under which the basic $k$-approximation for $H$-hitting set can be improved, advancing understanding of approximation limits for subgraph hitting problems.

Findings

Identifies classes of graphs $H$ where approximation factor can be improved.

Provides theoretical bounds for approximation ratios based on graph properties.

Highlights open problems in improving approximations for specific subgraph configurations.

Abstract

Let $H$ be a fixed undirected graph on $k$ vertices. The $H$-hitting set problem asks for deleting a minimum number of vertices from a given graph $G$ in such a way that the resulting graph has no copies of $H$ as a subgraph. This problem is a special case of the hypergraph vertex cover problem on $k$-uniform hypergraphs, and thus admits an efficient $k$-factor approximation algorithm. The purpose of this article is to investigate the question that for which graphs $H$ this trivial approximation factor $k$ can be improved.