Abstract
In conflict-directed clause learning (CDCL) SAT solving, a state-of-the-art criterion to measure the importance of a learned clause is called literal block distance (LBD), which is the number of distinct decision levels in the clause. The lower the LBD score of a learned clause, the better is its quality. The learned clauses with LBD score of 2, called glue clauses, are known to possess high pruning power. In this work, we relate glue clauses to decision variables. First, we show experimentally that branching decisions with variables appearing in glue clauses, called glue variables, are more conflict efficient than with nonglue variables. This observation motivated the development of a structure-aware CDCL variable bumping scheme, which increases the heuristic score of a glue variable based on its appearance count in the glue clauses that are learned so far by the search. Empirical evaluation shows the effectiveness of the new method on the main track instances from SAT Competitions 2017 and 2018 with four state-of-the-art CDCL SAT solvers. Finally, we show that the frequency of learned clauses that are glue clauses can be used as a reliable indicator of solving efficiency for some instances, for which the standard performance metrics fail to provide a consistent explanation.
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Notes
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A metric used in SAT competitions. Defined as the sum of all runtimes for solved instances + \(2*timeout\) for unsolved instances; lowest score wins.
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We omit the parameter s since the glue level of a variable is always computed w.r.t. a underlying search state by default, without confusion.
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At a given state of the search, a given glue variable v appears in at least one glue clause. So, the glue level of v (which is the count of number of glue clauses in which v appears), \(gl(v) > 0\). After dividing gl(v) with \(\vert G \vert \), the normalized glue level remains larger than 0. Hence, the normalization normalizes the glue level within the range (0,1].
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A total of 483 instances (283 applications, 200 crafted) after removing 17 duplicate instances between SAT-2016 and SAT-2017.
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Acknowledgements
We thank the anonymous reviewers for their valuable advice. This research is supported by Natural Sciences and Engineering Research Council of Canada (NSERC) PGS Doctoral award, President’s Doctoral Prize of Distinction (PDPD), Alberta Innovates Graduate Student Scholarship (AIGSS), and NSERC discovery grant.
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Chowdhury, M.S., Müller, M., You, JH. (2019). Exploiting Glue Clauses to Design Effective CDCL Branching Heuristics. In: Schiex, T., de Givry, S. (eds) Principles and Practice of Constraint Programming. CP 2019. Lecture Notes in Computer Science(), vol 11802. Springer, Cham. https://doi.org/10.1007/978-3-030-30048-7_8
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