MAX k-CUT and approximating the chromatic number of random graphs


  • An extended abstract version of this paper appeared in Proc. ICALP 2003, Springer LNCS 2719, pp. 200–211.


We consider the MAX k-CUT problem on random graphs Gn,p. First, we bound 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 in expected polynomial time, with approximation ratio 1 + O((np)-1/2). Our main technical tool is a new bound on the probable value of Frieze and Jerrum's semidefinite programming (SDP)-relaxation of MAX k-CUT on random graphs. To obtain this bound, we show that the value of the SDP is tightly concentrated. As a further application of our bound on the probable value of the SDP, we obtain an algorithm for approximating the chromatic number of Gn,p, 1/np ≤ 0.99, within a factor of O((np)1/2) in polynomial expected time, thereby answering a question of Krivelevich and Vu. We give similar algorithms for random regular graphs. The techniques for studying the SDP apply to a variety of SDP relaxations of further NP-hard problems on random structures and may therefore be of independent interest. For instance, to bound the SDP we estimate the eigenvalues of random graphs with given degree sequences. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006