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

E. M. Garzón, J. A. Martínez, J. J. Moreno, M. L. Puertas

TL;DR

The paper tackles computing the $2$-domination number in cylinders $P_m \Box C_n$ by leveraging the $(\min,+)$ matrix product (tropical algebra) on HPC platforms. It builds a state-encoding digraph $\mathcal{D}_m$ where $2$-dominating sets correspond to closed walks, and uses $(A(\mathcal{D}_m)^n)_{ii}$ to recover $\gamma_2$; a finite-difference recurrence further enables efficient handling of large $n$. The authors implement optimized $(\min,+)$-powered matrix routines on CPUs (OpenMP) and GPUs (CUDA), allowing $m$ up to $12$ to be explored and providing insights such as regularity for $m \ge 8$ and $n \equiv 0 \pmod{3}$, where $\gamma_2(P_m \Box C_n)$ matches the formula $\frac{(m+2)n}{3}$ in many cases. They also contribute open-source code and conjecture general patterns for large cylinders, highlighting the practical impact of HPC-accelerated tropical algebra in graph parameter computation.

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.

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

TL;DR

The paper tackles computing the -domination number in cylinders by leveraging the matrix product (tropical algebra) on HPC platforms. It builds a state-encoding digraph where -dominating sets correspond to closed walks, and uses to recover ; a finite-difference recurrence further enables efficient handling of large . The authors implement optimized -powered matrix routines on CPUs (OpenMP) and GPUs (CUDA), allowing up to to be explored and providing insights such as regularity for and , where matches the formula in many cases. They also contribute open-source code and conjecture general patterns for large cylinders, highlighting the practical impact of HPC-accelerated tropical algebra in graph parameter computation.

Abstract

The computation of the domination-type parameters is a challenging problem in Cartesian product graphs. We present an algorithmic method to compute the -domination number of the Cartesian product of a path with small order and any cycle, involving the 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 -domination number of cylinders such that their paths have at most vertices.
Paper Structure (6 sections, 4 theorems, 3 equations, 1 figure, 6 tables, 5 algorithms)

This paper contains 6 sections, 4 theorems, 3 equations, 1 figure, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. The 2-domination number in cylindrical graphs with small paths
  3. General construction
  4. Specific construction for the 2-domination number
  5. Algorithms and computational analysis
  6. Conclusions

Key Result

Theorem 1

Let $S_{ij}^n$ be the set of all paths of length $n$ from $v_i$ to $v_j$ in $\mathcal{D}$ and let $A(\mathcal{D})$ be the matrix defined by If $A(\mathcal{D})^n$ is the $n$-th $(\min, +)$ power of $A(\mathcal{D})$, then $(A(\mathcal{D})^n)_{ij}=\min \{\ell(Q)\colon Q\in S_{ij}^n\}.$

Figures (1)

  • Figure 1: A $2$-dominating set of $P_4\Box C_5$ and its associated word list.

Theorems & Definitions (5)

  • Theorem 1: Carre1979
  • Example 1
  • Theorem 2
  • Lemma 3
  • Theorem 4