From Horn Strong Backdoor Sets to Ordered Strong Backdoor Sets

Identifying and exploiting hidden problem structures is recognized as a fundamental way to deal with the intractability of combinatorial problems. Recently, a particular structure called (strong) backdoor has been identified in the context of the satisfiability problem. Connections has been established between backdoors and problem hardness leading to a better approximation of the worst case time complexity. Strong backdoor sets can be computed for any tractable class. In [1], a method for the approximation of strong backdoor sets for the Horn-Sat fragment was proposed. This approximation is realized in two steps. First, the best Horn renaming of the original CNF formula, in term of number of clauses, is computed. Then a Horn strong backdoor set is extracted from the non Horn part of the renamed formula. in this article, we propose computing Horn strong backdoor sets using the same scheme but minimizing the number of positive literals in the non Horn part of the renamed formula instead of minimizing the number of non Horn clauses. Then we extend this method to the class of ordered formulas [2] which is an extension of the Horn class. This method insure to obtain ordered strong backdoor sets of size less or equal than the size of Horn strong backdoor sets (never greater). Experimental results show that these new methods allow to reduce the size of strong backdoor sets on several instances and that their exploitation also allow to enhance the efficiency of satisfiability solvers.