Linear upper bounds for random walk on small density random 3-CNFs

We analyze the efficiency of the random walk algorithm on random 3-CNF instances, and prove linear upper bounds on the running time of this algorithm for small clause density, less than 1.63. Our upper bound matches the observed running time to within a multiplicative factor. This is the first sub-exponential upper bound on the running time of a local improvement algorithm on random instances. Our proof introduces a simple, yet powerful tool for analyzing such algorithms, which may be of further use. This object, called a terminator, is a weighted satisfying assignment. We show that any CNF having a good (small weight) terminator is assured to be solved quickly by the random walk algorithm. This raises the natural question of the terminator threshold which is the maximal clause density for which such assignments exist (with high probability). We use the analysis of the pure literal heuristic presented by Broder, Frieze and Upfal and show that for small clause densities good terminators exist. Thus we show that the pure literal threshold (≈ 1.63) is a lower bound on the terminator threshold. One nice property of terminators is that they can be found efficiently, via linear programming. This makes tractable the future investigation of the terminator threshold, and also provides an efficiently computable certificate for short running time of the simple random-walk heuristic.