An Improved Approximation Algorithm for Maximum Weight 3-Path Packing

TL;DR

This paper introduces a new $10/17$-approximation algorithm for the maximum weight 3-path packing problem in complete graphs, improving upon previous algorithms by combining multiple matching and star packing strategies.

Contribution

The paper presents a novel $10/17$-approximation algorithm that improves the approximation ratio for maximum weight 3-path packing, with a new analysis method applicable to related problems.

Findings

Achieved a better approximation ratio of 10/17 compared to previous 7/12.

Developed a new charging analysis method for approximation algorithms.

Combined multiple algorithms to optimize the solution.

Abstract

Given a complete graph with $n$ vertices and non-negative edge weights, where $n$ is divisible by 3, the maximum weight 3-path packing problem is to find a set of $n/3$ vertex-disjoint 3-paths such that the total weight of the 3-paths in the packing is maximized. This problem is closely related to the classic maximum weight matching problem. In this paper, we propose a $10/17$-approximation algorithm, improving the best-known $7/12$-approximation algorithm (ESA 2015). Our result is obtained by making a trade-off among three algorithms. The first is based on the maximum weight matching of size $n/2$, the second is based on the maximum weight matching of size $n/3$, and the last is based on an approximation algorithm for star packing. Our first algorithm is the same as the previous $7/12$-approximation algorithm, but we propose a new analysis method -- a charging method -- for this problem, which is not only essential to analyze our second algorithm but also may be extended to analyze algorithms for some related problems.