Hitting Diamonds and Growing Cacti

TL;DR

This paper introduces a constant-factor approximation algorithm for a complex NP-hard graph problem involving cycle disjointness, and analyzes the LP relaxation's integrality gap, revealing its logarithmic bound.

Contribution

It presents a primal-dual based approximation algorithm for the problem and establishes the LP relaxation's integrality gap as a( ext{log} n), a novel theoretical insight.

Findings

The algorithm achieves a constant-factor approximation.

The LP relaxation has an integrality gap of a( ext{log} n).

The problem is NP-hard and challenging to approximate.

Abstract

We consider the following NP-hard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constant-factor approximation algorithm, based on the primal-dual method. Moreover, we show that the integrality gap of the natural LP relaxation of the problem is \Theta(\log n), where n denotes the number of vertices in the graph.