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

TL;DR

This work tackles computing the Roman domination number $\gamma_R$ on cylindrical graphs $P_m\Box C_n$ by leveraging min-plus (tropical) matrix powers on GPUs. It builds a tropical-algebra framework with a digraph of correct words and a weighted adjacency matrix $A(G)$ whose powers yield minimal Roman-domination weights via closed walks; a finite-difference recurrence extends results beyond fixed $m$. The authors obtain exact formulas for $\gamma_R(P_7\Box C_n)$ and $\gamma_R(P_8\Box C_n)$ as functions of $n\bmod5$, and establish a general lower bound $\gamma_R(P_m\Box C_n) \ge \frac{2(m+1)n}{5}$ for $m,n\ge10$, which is tight when $n$ is a multiple of 5. They implement a GPU-accelerated routine CuMatrixTrop to perform the $(\min,+)$ powers efficiently and demonstrate the approach on CUDA hardware, highlighting the practical value of tropical-algebra techniques for graph parameters on large-scale structures. Overall, the paper extends exact results for cylinder graphs and showcases a scalable computational approach with potential for broader classes of Cartesian product graphs.

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 $γ_R(G)$ 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 $m\geq 10, n\not\equiv 0\pmod 5$.

Theorems & Definitions (14)

• Theorem 1
• Lemma 2
• Remark 3
• Definition 4
• Remark 5
• Proposition 6
• Corollary 7
• Proposition 8
• Definition 9
• Lemma 10