HPC acceleration of large (min, +) matrix products to compute domination-type parameters in graphs

TL;DR

This paper develops GPU-accelerated algorithms for computing the 2-domination number in Cartesian product graphs, specifically cylinders with small path sizes, improving computational efficiency over sequential methods.

Contribution

It introduces a novel algorithmic approach using $( ext{min},+)$ matrix products for large matrix computations in graph parameters, with performance analysis on modern hardware.

Findings

GPU implementation outperforms sequential algorithms

Able to compute 2-domination for cylinders with paths up to 12 vertices

Provides theoretical foundations for $( ext{min},+)$ matrix product applications in graph parameters

Abstract

The computation of the domination-type parameters is a challenging problem in Cartesian product graphs. We present an algorithmic method to compute the $2$-domination number of the Cartesian product of a path with small order and any cycle, involving the $(\min,+)$ matrix product. We establish some theoretical results that provide the algorithms necessary to compute that parameter, and the main challenge to run such algorithms comes from the large size of the matrices used, which makes it necessary to improve the techniques to handle these objects. We analyze the performance of the algorithms on modern multicore CPUs and on GPUs and we show the advantages over the sequential implementation. The use of these platforms allows us to compute the $2$-domination number of cylinders such that their paths have at most $12$ vertices.