Powers of Large Matrices on GPU Platforms to Compute the Roman Domination Number of Cylindrical Graphs

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