We show that a random instance of a weighted maximum constraint satisfaction problem (or MAX 2-CSP), whose clauses are over pairs of binary variables, is solvable by a deterministic algorithm in polynomial expected time, in the "sparse" regime where the expected number of clauses is half the number of variables. In particular, a maximum cut in a random graph with edge density 1/n or less can be found in polynomial expected time. Our method is to show, first, that if a MAX 2-CSP has a connected underlying graph with n vertices and m edges, the solution time can be deterministically bounded by 2(m-n)/2. Then, analyzing the tails of the distribution of this quantity for a component of a random graph yields our result. An alternative deterministic bound on the solution time, as 2m/5, improves upon a series of recent results. © Springer-Verlag Berlin Heidelberg 2003.
Scott, A. D., & Sorkin, G. B. (2003). Faster algorithms for MAX CUT and MAX CSP, with polynomial expected time for sparse instances. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2764, 382–395. https://doi.org/10.1007/978-3-540-45198-3_32