On decision and optimization (k, l)-graph sandwich problems

  • Authors:

  • Simone Dantas;Celina M. H. de Figueiredo;Luerbio Faria

  • Affiliations:

  • COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil;COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil and Instituto de Matemalica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro 21945-970, RJ, Brazil;Departamento de Matemática, Faculdade de Formação de Professores, UERJ, Brazil

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2004

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Abstract

A graph G is (k,l) if its vertex set can be partitioned into at most k independent sets and l cliques. The (k,l)-Graph Sandwich Problem asks, given two graphs G1 = (V,E1) and G2 = (V,E2), whether there exists a graph G = (V,E) such that E1 ⊆ E ⊆ E2 and G is (k,l). In this paper, we prove that the (k,l)-Graph Sandwich Problem is NP-complete for the cases k=1 and l=2; k=2 and l=1; or k=l=2. This completely classifies the complexity of the (k,l)-Graph Sandwich Problem as follows: the problem is NP-complete, if k+l 2; the problem is polynomial otherwise. We consider the degree Δ constraint subproblem and completely classify the problem as follows: the problem is polynomial, for k ≤ 2 or Δ ≤ 3; the problem is NP-complete otherwise. In addition, we propose two optimization versions of graph sandwich problem for a property Π: MAX-Π-GSP and MIN-Π-GSP. We prove that MIN-(2,1)-GSP is a Max-SNP-hard problem, i.e., there is a positive constant ε, such that the existence of an ε-approximative algorithm for MIN-(2,1)-GSP implies P = NP.