The complexity of counting homomorphisms seen from the other side

  • Authors:

  • Víctor Dalmau;Peter Jonsson

  • Affiliations:

  • Departament de tecnologia, Universitat Pompeu Fabra, Estació de França, Passeig de la circumval.lació 8, 08003, Barcelona, Spain;Department of Computer and Information Science, Linköpings Universitet, SE-581 83 Linköping, Sweden

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2004

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Abstract

For every class of relational structures C, let HOM(C, _) be the problem of deciding whether a structure A ∈ C has a homomorphism to a given arbitrary structure B. Grohe has proved that, under a certain complexity-theoretic assumption, HOM(C, _) is solvable in polynomial time if and only if the cores of all structures in C have bounded tree-width. We prove (under a weaker complexity-theoretic assumption) that the corresponding counting problem #HOM(C, _) is solvable in polynomial time if and only if all structures in C have bounded tree-width. This answers an open question posed by Grohe.