Random sampling of colourings of sparse random graphs with a constant number of colours

Authors:
Charilaos Efthymiou;Paul G. Spirakis
Affiliations:
Research Academic Computer Technology Institute, N. Kazantzaki str. Rio Patras, 26500, Greece and Computer Engineering and Informatics Department of the University of Patras, 26500, Greece;Research Academic Computer Technology Institute, N. Kazantzaki str. Rio Patras, 26500, Greece and Computer Engineering and Informatics Department of the University of Patras, 26500, Greece
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

In this work we present a simple and efficient algorithm which, with high probability, provides an almost uniform sample from the set of proper k-colourings on an instance of sparse random graphs G"n","d"/"n, where k=k(d) is a sufficiently large constant. Our algorithm is not based on the Markov Chain Monte Carlo method (M.C.M.C.). Instead, we provide a novel proof of correctness of our algorithm that is based on interesting ''spatial mixing'' properties of colourings of G"n","d"/"n. Our result improves upon previous results (based on M.C.M.C.) that required a number of colours growing unboundedly with n.