Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
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The Boolean Satisfiability problem asks if a Boolean formula is satisfiable by some assignment of the variables or not. It belongs to the NP-complete complexity class and hence no algorithm with polynomial time worst-case complexity is known, i.e., the problem is hard. The K-SAT problem is the subset of the Boolean Satisfiability problem, for which the Boolean formula has the conjunctive normal form with K literals per clause. This problem is still NP-complete for K > 3. Although the worst case complexity of NP-complete problems is conjectured to be exponential, there might be subsets of the realizations where solutions can typically be found in polynomial time. In fact, random K-SAT, with the number of clauses to number of variables ratio α as control parameter, shows a phase transition between a satisfiable phase and an unsatisfiable phase, at which the hardest problems are located. We use here several linear programming approaches to reveal further "easy-hard" transition points at which the typical hardness of the problems increases which means that such algorithms can solve the problem on one side efficiently but not beyond this point. For one of these transitions, we observed a coincidence with a structural transition of the literal factor graphs of the problem instances. We also investigated cutting-plane approaches, which often increase the computational efficiency. Also we tried out a mapping to another NP-complete optimization problem using a specific algorithm for that problem. In both cases, no improvement of the performance was observed, i.e., no shift of the easy-hard transition to higher values of α.
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