Abstract
We revisit in this paper the stochastic model for minimum graph-coloring introduced in (Murat and Paschos in Discrete Appl. Math. 154:564–586, 2006), and study the underlying combinatorial optimization problem (called probabilistic coloring) in bipartite and split graphs. We show that the obvious 2-coloring of any connected bipartite graph achieves standard-approximation ratio 2, that when vertex-probabilities are constant probabilistic coloring is polynomial and, finally, we propose a polynomial algorithm achieving standard-approximation ratio 8/7. We also handle the case of split graphs. We show that probabilistic coloring is NP-hard, even under identical vertex-probabilities, that it is approximable by a polynomial time standard-approximation schema but existence of a fully a polynomial time standard-approximation schema is impossible, even for identical vertex-probabilities, unless P=NP. We finally study differential-approximation of probabilistic coloring in both bipartite and split graphs.
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Part of this research has been performed while the second author was with the LAMSADE on a research position funded by the CNRS.
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Bourgeois, N., Della Croce, F., Escoffier, B. et al. Probabilistic graph-coloring in bipartite and split graphs. J Comb Optim 17, 274–311 (2009). https://doi.org/10.1007/s10878-007-9112-2
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DOI: https://doi.org/10.1007/s10878-007-9112-2