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 :V (G) -> {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) = Sigma(u is an element of 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 square C-n i.e., the Cartesian product of a path and a cycle, in cases m = 7, 8,9 n >= 3 and m >= 10, n equivalent to 0 (mod 5). Moreover, we provide a lower bound for the remaining cases m >= 10, n (sic) 0 (mod 5).