Approximately covering vertices by order-$5$ or longer paths

TL;DR

This paper introduces a new approximation algorithm for covering vertices with long paths in a graph, improving the ratio from 2.714 to 2.511 and using advanced matching and cover techniques.

Contribution

The paper presents a novel approximation algorithm for the vertex covering problem with long paths, improving the approximation ratio and computational efficiency.

Findings

Achieves an approximation ratio of 2.511 for the problem.

Runs in O(|V|^{2.5}|E|^2) time, faster than previous algorithms.

Utilizes maximum matching, path-cycle cover, and recursion techniques.

Abstract

This paper studies $MPC^{5+}_v$, which is to cover as many vertices as possible in a given graph $G=(V,E)$ by vertex-disjoint $5^+$-paths (i.e., paths each with at least five vertices). $MPC^{5+}_v$ is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of $\frac {19}7\approx 2.714$ and runs in $O(|V|^6)$ time. In this paper, we present a new approximation algorithm for $MPC^{5+}_v$ which achieves a ratio of $2.511$ and runs in $O(|V|^{2.5} |E|^2)$ time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion.