Backbone Fragility Causes the Local Search Cost Peak

The local search algorithm Wsat is one of the most successful algorithms for solving the satis ability (SAT) problem. It is notably e ective at solving hard Random 3-SAT instances near the so-called `satis ability threshold', but still shows a peak in search cost near the threshold and large variations in cost over di erent instances. We make a number of signi cant contributions to the analysis ofWsat on high-cost random instances, using the recent 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 in Wsat'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. This explains the decay in cost beyond the satis ability threshold as being due to increasing backbone robustness. Our hypothesis makes three correct predictions. First, that the backbone robustness of an instance is negatively correlated with the local search cost when other factors are controlled for. Second, that backbone-minimal instances (which are 3-SAT instances altered so as to be more backbone-fragile) are unusually hard forWsat. Third, that the clauses most often unsatis ed during search are those whose deletion has the most e ect on the backbone. In understanding the pathologies of local search methods, we hope to contribute to the development of new and better techniques.