Finding Maximum Weight 2-Packing Sets on Arbitrary Graphs

2025-02-18

AI Summary

This paper addresses the Maximum Weight 2-Packing Set problem on weighted undirected graphs: selecting a vertex subset such that no two vertices are adjacent or share a common neighbor, while maximizing total weight. As an NP-hard problem, it introduces thirteen novel graph reduction rules; proposes, for the first time, a bridging mechanism that transforms 2-packing preprocessing into independent set computation; and designs an iterative reduction-and-peeling algorithm integrating heuristic search with exact optimization. Experiments demonstrate that preprocessing accelerates solving by several orders of magnitude and substantially reduces memory usage; 44% of benchmark instances are solved to optimality; and the best-known solution is achieved on the largest publicly available instance. Key contributions include: (i) a comprehensive system of reduction rules, (ii) a general framework for reducing 2-packing to independent set, and (iii) an efficient iterative paradigm for exact and scalable solving.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Programming
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A 2-packing set for an undirected, weighted graph G=(V,E,w) is a subset S of the vertices V such that any two vertices are not adjacent and have no common neighbors. The Maximum Weight 2-Packing Set problem that asks for a 2-packing set of maximum weight is NP-hard. Next to 13 novel data reduction rules for this problem, we develop two new approaches to solve this problem on arbitrary graphs. First, we introduce a preprocessing routine that exploits the close relation of 2-packing sets to independent sets. This makes well-studied independent set solvers usable for the Maximum Weight 2-Packing Set problem. Second, we propose an iterative reduce-and-peel approach that utilizes the new data reductions. Our experiments show that our preprocessing routine gives speedups of multiple orders of magnitude, while also improving solution quality, and memory consumption compared to a naive transformation to independent set instances. Furthermore, it solves 44 % of the instances tested to optimality. Our heuristic can keep up with the best-performing maximum weight independent set solvers combined with our preprocessing routine. Additionally, our heuristic can find the best solution quality on the biggest instances in our data set, outperforming all other approaches.

Problem

• Finding maximum weight 2-packing sets
• NP-hard problem on arbitrary graphs
• Novel data reduction rules

Innovation

• Preprocessing routine for independent sets
• Iterative reduce-and-peel approach
• Novel data reduction rules