An Improved Approximation Algorithm for Maximum Weight 3-Path Packing

2025-12-16

AI Summary

For the Maximum-Weight 3-Path Packing problem on a complete graph—i.e., packing (n/3) vertex-disjoint 3-paths (paths of three vertices) on (n) vertices ((n equiv 0 pmod{3})) to maximize total weight—we present the first (10/17)-approximation algorithm, improving upon the previous best ratio of (7/12). Our method integrates maximum-weight matchings of sizes (n/2) and (n/3), an approximation for star packing, and a novel “charging-based” analysis technique. Theoretically, we introduce a unified analytical framework grounded in the “assignment method”, which systematically characterizes the performance guarantees of three distinct subroutine classes for the first time. This framework not only yields the improved approximation ratio but also generalizes naturally to broader path-packing and structural packing problems. Beyond this specific result, our framework establishes a transferable analytical paradigm for combinatorial optimization problems involving disjoint subgraph packing.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social media analysis through the lenses of networks

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.

Problem

• Develops a 10/17-approximation algorithm for maximum weight 3-path packing
• Improves upon the previous best 7/12-approximation algorithm
• Uses a trade-off among three algorithms with a new charging analysis method

Innovation

• Combining three algorithms for improved approximation ratio
• Using charging method for novel analysis of algorithm performance
• Focusing on maximum weight matching and star packing techniques