Algorithms for quantified Boolean formulas

  • Authors:

  • Ryan Williams

  • Affiliations:

  • Cornell University, Ithaca, NY

  • Venue:
  • SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2002

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Abstract

We present algorithms for solving quantified Boolean formulas (QBF, or sometimes QSAT) with worst case runtime asymptotically less than O(2n) when the clause-to-variable ratio is smaller or larger than some constant. We solve QBFs in conjunctive normal form (CNF) in O(1.709m) time and space, where m is the number of clauses. Extending the technique to a quantified version of constraint satisfaction problems (QCSP), we solve QCSP with domain size d = 3 in O(1.953m) time, and QCSPs with d ≥ 4 in O(dm/2+ε) time and space for ε m is the number of constraints. For 3-CNF QBF, we describe an polynomial space algorithm with time complexity O(1.619n) when the number of 3-CNF clauses is equal to n; the bound approaches 2n as the clause-to-variable ratio approaches 2. For 3-CNF Π2-SAT (3-CNF QBFs of the form ∀u1…uj∃xj+1…xnF), an improved polyspace algorithm has runtime varying from O(1.840m) to O(1.415m), as a particular clause-to-variable ratio increases from 1.