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NP by Means of Lifts and Shadows

  • Gábor Kun
  • Jaroslav Nešetřil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden colored (lifted) subgraphs. Our characterization is motivated by the analysis of syntactical subclasses with the full computational power of NP, which were first studied by Feder and Vardi.

Our approach applies to many combinatorial problems and it induces the characterization of coloring problems (CSP) defined by means of shadows. This turns out to be related to homomorphism dualities. We prove that a class of digraphs (relational structures) defined by finitely many forbidden colored subgraphs (i.e. lifted substructures) is a CSP class if and only if all the the forbidden structures are homomorphically equivalent to trees. We show a surprising richness of coloring problems when restricted to most frequent graph classes. Using results of Nešetřil and Ossona de Mendez for bounded expansion classes (which include bounded degree and proper minor closed classes) we prove that the restriction of every class defined as the shadow of finitely many colored subgraphs equals to the restriction of a coloring (CSP) class.

Keywords

Digraph homomorphism duality NP Constraint Satisfaction Problem 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gábor Kun
    • 1
  • Jaroslav Nešetřil
    • 1
  1. 1.Department of Mathematics, University of Memphis, 373 Dunn Hall, Memphis, TN 38152, Department of Applied Mathematics (KAM) and, Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám 22, Praha 

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