Minimum Cost Homomorphism Dichotomy for Oriented Cycles

Authors:
Gregory Gutin;Arash Rafiey;Anders Yeo
Affiliations:
Department of Computer Science Royal Holloway, University of London, Egham, UK TW20 0EX;School of Computing Science, Simon Fraser University, Burnaby, Canada V5A 1S6;Department of Computer Science Royal Holloway, University of London, Egham, UK TW20 0EX
Venue:
AAIM '08 Proceedings of the 4th international conference on Algorithmic Aspects in Information and Management
Year:
2008

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Abstract

For digraphs Dand H, a mapping f: V(D) 驴V(H) is a homomorphism of Dto Hif uv驴 A(D) implies f(u)f(v) 驴 A(H). If, moreover, each vertex u驴 V(D) is associated with costs ci(u), i驴 V(H), then the cost of the homomorphism fis 驴 u驴 V(D)cf(u)(u). For each fixed digraph H, we have the minimum cost homomorphism problem forH(abbreviated MinHOM(H)). In this discrete optimization problem, we are to decide, for an input graph Dwith costs ci(u), u驴 V(D), i驴 V(H), whether there exists a homomorphism of Dto Hand, if one exists, to find one of minimum cost. We obtain a dichotomy classification for the time complexity of MinHOM(H) when His an oriented cycle. We conjecture a dichotomy classification for all digraphs with possible loops.