A dichotomy for minimum cost graph homomorphisms

  • Authors:

  • Gregory Gutin;Pavol Hell;Arash Rafiey;Anders Yeo

  • Affiliations:

  • Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK and Department of Computer Science, University of Haifa, Israel;School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A 1S6;School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, V5A 1S6;Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, UK

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2008

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Abstract

For graphs G and H, a mapping f:V(G)-V(H) is a homomorphism of G to H if uv@?E(G) implies f(u)f(v)@?E(H). If, moreover, each vertex u@?V(G) is associated with costs c"i(u),i@?V(H), then the cost of the homomorphism f is @?"u"@?"V"("G")c"f"("u")(u). For each fixed graph H, we have the minimum cost homomorphism problem, written as MinHOM(H). The problem is to decide, for an input graph G with costs c"i(u),u@?V(G),i@?V(H), whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H, with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM(H) is polynomial time solvable. In all other cases the problem MinHOM(H) is NP-hard. This solves an open problem from an earlier paper.