Powers of large matrices on GPU platforms to compute the Roman domination number of cylindrical graphs

TL;DR

This paper develops GPU-accelerated algorithms to efficiently compute the Roman domination number of cylindrical graphs, leveraging large matrix powers, and provides results for specific graph sizes with bounds for others.

Contribution

It introduces novel GPU-based algorithms for calculating the Roman domination number of cylindrical graphs using $( ext{min},+)$ matrix powers, enabling analysis of larger graphs.

Findings

Successfully computed Roman domination numbers for certain cylindrical graphs.

Developed efficient GPU routines for $( ext{min},+)$ matrix multiplication.

Provided lower bounds for remaining graph cases.

Abstract

The Roman domination in a graph $G$ is a variant of the classical domination, defined by means of a so-called Roman domination function $f\colon V(G)\to \{0,1,2\}$ such that if $f(v)=0$ then, the vertex $v$ is adjacent to at least one vertex $w$ with $f(w)=2$. The weight $f(G)$ of a Roman dominating function of $G$ is the sum of the weights of all vertices of $G$, that is, $f(G)=\sum_{u\in V(G)}f(u)$. The Roman domination number $\gamma_R(G)$ is the minimum weight of a Roman dominating function of $G$. In this paper we propose algorithms to compute this parameter involving the $(\min,+)$ powers of large matrices with high computational requirements and the GPU (Graphics Processing Unit) allows us to accelerate such operations. Specific routines have been developed to efficiently compute the $(\min ,+)$ product on GPU architecture, taking advantage of its computational power. These algorithms allow us to compute the Roman domination number of cylindrical graphs $P_m\Box C_n$ i.e., the Cartesian product of a path and a cycle, in cases $m=7,8,9$, $ n\geq 3$ and $m\geq $10$, n\equiv 0\pmod 5$. Moreover, we provide a lower bound for the remaining cases $m\geq 10, n\not\equiv 0\pmod 5$.