Data reductions, fixed parameter tractability, and random weighted d-CNF satisfiability

摘要

Data reduction is a key technique in the study of fixed parameter algorithms. In the AI literature, pruning techniques based on simple and efficient-to-implement reduction rules also play a crucial role in the success of many industrial-strength solvers. Understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly-generated instances of NP-complete problems.In this paper, we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem, the weighted d-CNF satisfiability problem. We propose a non-trivial random model for the problem and study the probabilistic behavior of the random instances from the model. We design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed-parameter polynomial time O(dknm) where n is the number of variables, m is the number of clauses, and k is the fixed parameter. We establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space, unsatisfiable random instances of the problem have parametric resolution proof of fixed-parameter polynomial size. Also discussed is a more general random model and the generalization of the results to the model.