Solving Sparse Semi-Random Instances of Max Cut and Max CSP

We show that a maximum cut of a random graph below the giant-component threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable-constraint satisfaction problems, or Max 2-CSPs. In addition to Max Cut, Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum independent set, and minimum vertex cover. We show that if a Max 2-CSP instance has an “underlying” graph which is a random graph G(n, c/n), then the instance is solved in expected linear time if c 1. Moreover, for arbitrary values (or functions) c > 1 an instance is solved in expected time n exp(O(1 + (c - 1)³/n)); in the “scaling window” with fixed, this expected time remains linear.

Our method is to show, first, that if a Max 2-CSP has a connected underlying graph with n vertices and m edges, then nO(2^(m-n)/2) is a deterministic upper bound on the solution time. Then, analyzing the tails of the distribution of this quantity for a component of a random graph yields our result. Towards this end we derive some useful properties of binomial distributions and simple random walks.