Dynamic Variable Filtering for Hard Random 3-SAT Problems

The DPL (Davis-Putnam-Logemann-Loveland) procedure is one of the most effective methods for solving SAT problems. It is well known that its efficiency depends on the choice of the branching rule. One of the main improvements of this decision procedure has been the development of better branching variable selection through the use of unit propagation look-ahead (UPLA) heuristics (e.g., [12]). UPLA heuristics perform one of the following actions during two propagations of a free variable at each search tree node: detecting a contradiction earlier, simplifying the formula, or weighing the branching variable candidates. UPLA heuristics can be considered as polynomial time reasoning techniques. In this paper, we propose a new branching rule which does more reasoning to narrow the search tree of hard random 3-SAT problems. We term this reasoning technique the Dynamic Variable Filtering (DVF) heuristic. In our empirical study we develop four different variants of the DPL procedure: two (ssc34 and ssc355) based on this new heuristic and another two (Satz215-0 and Satz215sT) based on static variable filtering heuristics. ssc355 additionally integrates the Neighborhood Variable Ordering (NVO) heuristic into ssc34. We then compare the best known versions of the state-of-the-art Satz DPL procedure (Satz215), with each of our four procedures. Our results suggest that improved branching rules can further enhance the performance of DPL procedures. On hard random 3-SAT problems, our best procedure (ssc355) outperforms Satz215 with an order of magnitude in terms of the number of branching nodes in the search tree. While the run-times for dynamic variable filtering are still uncompetitive with Satz215, we have yet to explore the benefits that can be gained from avoiding redundant propagations and we still can improve the performance of the NVO heuristic. A further interesting property of dynamic filtering is that all backtracking can be eliminated during the DPL unit rule process. This property can also be explored in our future work for improving DPL procedure efficiency.

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