Finding Maximum Weight 2-Packing Sets on Arbitrary Graphs

TL;DR

This paper introduces novel data reduction rules and approaches for solving the NP-hard Maximum Weight 2-Packing Set problem on arbitrary graphs, significantly improving speed and solution quality.

Contribution

It presents 13 new data reduction rules, a preprocessing routine linking 2-packing to independent sets, and an iterative reduce-and-peel heuristic, advancing solution methods for this problem.

Findings

Preprocessing yields multi-order speedups and better solutions.

Solves 44% of instances to optimality.

Heuristic matches top independent set solvers and outperforms others on large instances.

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.