Um algoritmo de planos de corte para o problema livre de garra: estudo poliédrico e implementação Branch-and-Cut

Let G = (V;E) be a simple graph, where V is the set of vertices and E is the set of edges. A claw is de ned as a complete bipartite graph K1,3. The Claw-free problem (CFP) aims at nding a minimum subset of vertices S c V such that none of the vertex-induced subgraphs of G[V n S] are claws. The present work performs a polyhedral study on the CFP, which is a NP-complete problem, presenting 2 integer programming models(Fg and FSk ), where FSk is implemented with a cutting plane-based procedure. Computational experiments were performed in instances with di erent densities with up to 100 vertices. The results obtained suggest that FSk had a superior performance when compared to Fg.NenhumaSeja G = (V;E) um grafo, no qual V e o conjunto de vertices e E o conjunto de arestas. Um grafo garra é de nido como sendo um grafo bipartido completo K1,3. O Problema Livre de Garra (PLG) tem como objetivo encontrar um subconjunto minimo de vértices S c V de modo que nenhum subgrafo induzido por vértice em G[V / S] seja um grafo garra. O presente trabalho realiza um estudo poliedral para o PLG, que é um problema NP-completo, apresentando dois modelos de programa c~ao linear inteira (Fg e FSk ), sendo o ultimo implementado por meio de um procedimento baseado em planos de corte. Experimentos computacionais foram realizados em instâncias com diversas densidades e contendo até 100 vértices. Os resultados obtidos sugerem que FSk teve desempenho superior quando comparado a Fg

Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems