Lower Bounds for Myopic DPLL Algorithms with a Cut Heuristic

The paper is devoted to lower bounds on the time complexity of DPLL algorithms that solve the satisfiability problem using a splitting strategy. Exponential lower bounds on the running time of DPLL algorithms on unsatisfiable formulas follow from the lower bounds for resolution proofs. Lower bounds on satisfiable instances are also known for some classes of DPLL algorithms; this lower bounds are usually based on reductions to unsatisfiable instances. In this paper we consider DPLL algorithms with a cut heuristic that may decide that some branch of the splitting tree will not be investigated. DPLL algorithms with a cut heuristic always return correct answer on unsatisfiable formulas while they may err on satisfiable instances. We prove the theorem about effectiveness vs. correctness trade-off for deterministic myopic DPLL algorithms with a cut heuristic. Myopic algorithms can see formulas with erased signs of negations; they may also request a small number of clauses to read them precisely. We construct a family of unsatisfiable formulas Φ(n) and for every deterministic myopic algorithm A we construct polynomial time samplable ensemble of distributions Rn concentrated on satisfiable formulas such that if A gives a correct answer with probability 0.01 on a random formula from the ensemble Rn then A runs exponential time on the formulas Φ(n). One of the consequences of our result is the existence of a polynomial time samplable ensemble of distributions Qn concentrated on satisfiable formulas such that every deterministic myopic algorithm that gives a correct answer with probability 1− o(1) on a random formula from the ensemble Qn runs exponential time on the formulas Φ(n). Other consequence is the following statement: for every deterministic myopic algorithm A we construct a family of satisfiable formulas Ψ(n) and polynomial time samplable ensemble of distributions Rn concentrated on satisfiable formulas such that if A gives a correct answer with probability 0.01 on a random formula from the ensemble Rn then A runs exponential time on the formulas Ψ (n). ∗Preliminary version appeared in ISAAC 2011[IS11b]. †Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St.Petersburg, 191023, Russia, [email protected], [email protected]. The work is partially supported by Federal Target Programme “Scientific and scientific-pedagogical personnel of the innovative Russia” 2009-2013, the president grants MK-4108.2012.1, by RFBR grants 12-01-31239 mol a and 11-01-12135-ofi-m-2011 and by RAS Program for Fundamental Research.