Scalable Kernelization for Maximum Independent Sets

TL;DR

This paper introduces a fast, parallel kernelization algorithm for maximum independent sets that produces significantly smaller kernels and accelerates existing heuristics, improving efficiency on large graphs.

Contribution

The authors develop a parallel kernelization method combining graph partitioning, bipartite matching, and pruning techniques, achieving smaller kernels faster than previous methods.

Findings

Produces kernels orders of magnitude smaller than existing methods.

Achieves kernel computation up to two orders of magnitude faster.

Accelerates heuristic algorithms to find larger independent sets on large networks.

Abstract

The most efficient algorithms for finding maximum independent sets in both theory and practice use reduction rules to obtain a much smaller problem instance called a kernel. The kernel can then be solved quickly using exact or heuristic algorithms---or by repeatedly kernelizing recursively in the branch-and-reduce paradigm. It is of critical importance for these algorithms that kernelization is fast and returns a small kernel. Current algorithms are either slow but produce a small kernel, or fast and give a large kernel. We attempt to accomplish both of these goals simultaneously, by giving an efficient parallel kernelization algorithm based on graph partitioning and parallel bipartite maximum matching. We combine our parallelization techniques with two techniques to accelerate kernelization further: dependency checking that prunes reductions that cannot be applied, and reduction tracking that allows us to stop kernelization when reductions become less fruitful. Our algorithm produces kernels that are orders of magnitude smaller than the fastest kernelization methods, while having a similar execution time. Furthermore, our algorithm is able to compute kernels with size comparable to the smallest known kernels, but up to two orders of magnitude faster than previously possible. Finally, we show that our kernelization algorithm can be used to accelerate existing state-of-the-art heuristic algorithms, allowing us to find larger independent sets faster on large real-world networks and synthetic instances.