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Polynomial iterative algorithms for coloring and analyzing random graphs

A. Braunstein·R. Mulet·A. Pagnani·M. Weigt·R. Zecchina
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Abstract

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This lead us to propose a new algorithm able to color in polynomial time random graphs in the hard but colorable region, i.e when $c\in [c_d,c_q]$.

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A. Braunstein, R. Mulet, A. Pagnani, M. Weigt, R. Zecchina. 2003-04-24. Polynomial iterative algorithms for coloring and analyzing random graphs. https://doi.org/10.1103/physreve.68.036702

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