A note on the NP-hardness of the separation problem on some valid inequalities for the elementary shortest path problem

• Autor(a) principal: Ibrahim,M.S.
• Data de Publicação: 2014
• Outros Autores: Maculan,N., Minoux,M.
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Pesquisa operacional (Online)
• Texto Completo: http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0101-74382014000100009
• topic: polytope; digraphs; shortest path; valid inequality; separation

In this paper, we investigate the separation problem on some valid inequalities for the s - t elementary shortest path problem in digraphs containing negative directed cycles. As we will see, these inequalities depend to a given parameter k ∈ ℕ. To show the NP-hardness of the separation problem of these valid inequalities, considering the parameter k ∈ ℕ, we establish a polynomial reduction from the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s1, t1), (s2, t2) ... (sk+2, t k+2) in a digraph to the decision problem associated to the separation of these valid inequalities. Through some illustrative instances, we exhibit the evoked polynomial reduction in the cases k = 0 and k = 1.