Backbone Fragility and the Local Search Cost Peak

The local search algorithm WSat is one of the most successful algorithms for solving the satisability (SAT) problem. It is notably eective at solving hard Random 3-SAT instances near the so-called `satisability threshold', but still shows a peak in search cost near the threshold and large variations in cost over dierent instances. We make a number of signicant contributions to the analysis of WSat on high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are entailed by an instance. We nd that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We show a very strong correlation between search cost and the Hamming distance to the nearest solution early inWSat's search. This pattern leads us to introduce a measure of the backbone fragility of an instance, which indicates how persistent the backbone is as clauses are removed. We propose that high-cost random instances for local search are those with very large backbones which are also backbone-fragile. We suggest that the decay in cos