A proposta deste trabalho consiste em uma solução aproximada para o problema do subgrafo planar de peso máximo (WMPG Weighted Maximal Planar Graph). O algoritmo baseia-se na adição de vértices, aproveitando-se da construção de triangulações nas faces do grafo. A vantagem do uso deste algoritmo dá-se pelo fato que todo grafo gerado por ele é maximal planar, descartando a necessidade de um teste de planaridade. Apresentamos um algoritmo sequencial e um paralelo para o problema WMPG e suas respectivas implementações. Os resultados obtidos com a versão paralela executando em uma arquitetura manycore, com instâncias de até 85 vértices, apresentaram speedups de até 107 vezes em relação à solução sequencial.
ABSTRACT. Matrices are one of the most common representations of graphs. They are also used for representing algebras and cluster algebras. A symmetrizable matrix M is one for which there is a diagonal matrix D with positive entries, called symmetrizer matrix, such that DM is symmetric. This paper provides some properties of matrices in order to facilitate the understanding of symmetrizable matrices with specific characteristics, called positive quasi-Cartan companion matrices, and the problem of localizing them. We sharpen known coefficient limits for such matrices. By generalizing Sylvester's criterion for symmetrizable matrices we show that the localization problem is in NP and conjectured that it is NP-complete.
Network science is a growing field of study using Graph Theory as a modeling tool. In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. In this sense, a role assignment permit to represent the network through a smaller graph modeling its roles. This leads to a problem called r -Role Assignment whose goal is deciding whether it exists such an assignment of r distinct roles. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is a well studied graph operation, often used for modeling interconnection networks. Formally, the Cartesian product of G and H is a graph, denoted as G□H, whose vertex set is V (G) × V (H) and two vertices (u, v) and (x, y) are adjacent precisely if u = x and vy ∈ E(H), or ux ∈ E(G) and v = y. In a previous work, we showed that Cartesian product of graphs are always 2-role assignable, however the 3-Role Assignment problem is NP-complete on this class. In this paper, we prove that r -Role Assignment restricted to Cartesian product graphs is still NP-complete, for any fixed r ≥ 4.
In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. As a simple graph models a social network role assignment rises to the decision problem called r -Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r -Role Assignment restricted to Cartesian product of graphs, for r = 2,3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.
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