Branching rules for satisfiability

Abstract

Recent experience suggests that branching algorithms are among the most attractive for solving propositional satis ability problems. A k ey factor in their success is the rule they use to decide on which v ariable to branch next. We attempt to explain and improve the performance of branching rules with an empirical model-building approach. One model is based on the rationale given for the Jeroslow-Wang rule, variations of which have performed well in recent w ork. The model is refuted by carefully designed computational experiments. A second model explains the success of the Jeroslow-Wang rule, makes other predictions con rmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang.Recent computational studies 2, 7, 13, 21] suggest that branching algorithms are among the most attractive for solving the propositional satis ability problem. An important factor in their success|perhaps the dominant factor|is the branching rule they use 13]. This is a rule that decides, at each node of an enumeration tree, which variables should be set to true or false in order to generate the children of that node. A clever branching rule can reduce the size of the search tree by several orders of magnitude.One rule that has been found to be particularly e ective in a wide variety o f problems 13] is the Jeroslow-Wang rule 17], which w e de ne below. Another promising rule is the shortest positive clause rule used by Gallo and Urbani in their Horn relaxation algorithm 11]. There is little understanding, however, of when and why these rules work well.Our purpose here is to try to improve our understanding of branching rules and to design better ones. We will show that the original motivation for the J-W rule, namely that it takes a branch in which one is most likely to nd a satisfying truth assignment, does not explain its performance. A proper explanation is considerably more nuanced and reveals that the original motivation produces a good rule only through a remarkable coincidence.GSIA Working Paper 1994-09. The rst author is partially supported by ONR grant N00014-92-J-1028. The authors wish to thank Ajai Kapoor for assistance in computational testing and statistical analysis.

The Multi-Mode Resource-Constrained Project Scheduling Problem

“…The selection of the branching literals is an important factor for the efficiency (Hooker and Vinay, 1995). Obviously, infeasible instances and instances with tight non-renewable resource constraints might consume a lot of CPU time using the activity list discussed previously, since this list does not contain guiding information to select variables to branch.…”

Guiding CNF-SAT Search by Analyzing Constraint-Variable Dependencies and Clause Lengths

“…172-173). However, [19] has experimentally disproved their original justification and shown that the reason behind the success of JW heuristic is that it creates simpler sub-problems. This, in turn, leads to faster SAT solving.…”

Restoring Satisfiability or Maintaining Unsatisfiability by finding small Unsatisfiable Subformulae

“…Our SAT solution algorithm uses a clause-based search tree [13,10]. At every iteration, a clause C s to be satisfied is selected.…”

“…Priority is given to unit clauses (unit resolution). After them, since starting assignment by satisfying the more difficult clauses is known to be very helpful in reducing backtracks [10], we select hardest clauses. Hardness of a clause is evaluated by means of the below ϕ(C j ).…”