Abstract
We consider the MAX k-CUT problem in random graphs G n,p. First, we estimate the probable weight of a MAX k-CUT using probabilistic counting arguments and by analyzing a simple greedy heuristic. Then, we give an algorithm that approximates MAX k-CUT within expected polynomial time. The approximation ratio tends to 1 as np→ ∞. As an application, we obtain an algorithm for approximating the chromatic number of G n,p, 1/n≤ p ≤ 1/2, within a factor of \( O\left( {\sqrt {np} } \right) \) in polynomial expected time, thereby answering a question of Krivelevich and Vu, and extending a result of Coja-Oghlan and Taraz. We give similar algorithms for random regular graphs G n,r.
Research supported by the Deutsche Forschungsgemeinschaft (DFG FOR 413/1-1)
Supported by NSF PHY-0071139 and Los Alamos National Laboratory.
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Coja-Oghlan, A., Moore, C., Sanwalani, V. (2003). MAX k-CUT and Approximating the Chromatic Number of Random Graphs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_18
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