Scalable Algorithms for 2-Packing Sets on Arbitrary Graphs

TL;DR

This paper introduces a new algorithm for the NP-hard 2-packing set problem in graphs, leveraging data reduction and graph transformation to significantly improve solution quality and speed over existing methods.

Contribution

The authors develop a novel approach with specific data reduction rules and transformations, enabling faster and more effective solutions for 2-packing sets on arbitrary graphs.

Findings

Outperforms state-of-the-art in solution quality and speed

Solves 63% of test graphs to optimality in under a second

Handles large, previously unsolvable instances

Abstract

A 2-packing set for an undirected graph $G=(V,E)$ is a subset $\mathcal{S} \subset V$ such that any two vertices $v_1,v_2 \in \mathcal{S}$ have no common neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem. We develop a new approach to solve this problem on arbitrary graphs using its close relation to the independent set problem. Thereby, our algorithm red2pack uses new data reduction rules specific to the 2-packing set problem as well as a graph transformation. Our experiments show that we outperform the state-of-the-art for arbitrary graphs with respect to solution quality and also are able to compute solutions multiple orders of magnitude faster than previously possible. For example, we are able to solve 63% of the graphs in the tested data set to optimality in less than a second while the competitor for arbitrary graphs can only solve 5% of these graphs to optimality even with a 10 hour time limit. Moreover, our approach can solve a wide range of large instances that have previously been unsolved.