Hard Satisfiable Formulas for Splittings by Linear Combinations

  • Dmitry Itsykson
  • Alexander Knop
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)


Itsykson and Sokolov in 2014 introduced the class of \(\mathrm {DPLL}(\oplus )\) algorithms that solve Boolean satisfiability problem using the splitting by linear combinations of variables modulo 2. This class extends the class of \(\mathrm {DPLL}\) algorithms that split by variables. \(\mathrm {DPLL}(\oplus )\) algorithms solve in polynomial time systems of linear equations modulo 2 that are hard for \(\mathrm {DPLL}\), \(\mathrm {PPSZ}\) and \(\mathrm {CDCL}\) algorithms. Itsykson and Sokolov have proved first exponential lower bounds for \(\mathrm {DPLL}(\oplus )\) algorithms on unsatisfiable formulas.

In this paper we consider a subclass of \(\mathrm {DPLL}(\oplus )\) algorithms that arbitrary choose a linear form for splitting and randomly (with equal probabilities) choose a value to investigate first; we call such algorithms drunken \(\mathrm {DPLL}(\oplus )\). We give a construction of a family of satisfiable CNF formulas \(\varPsi _n\) of size \(\mathrm {poly}(n)\) such that any drunken \(\mathrm {DPLL}(\oplus )\) algorithm with probability at least \(1 - 2^{-\varOmega (n)}\) runs at least \(2^{\varOmega (n)}\) steps on \(\varPsi _n\); thus we solve an open question stated in the paper [12]. This lower bound extends the result of Alekhnovich, Hirsch and Itsykson [1] from drunken \(\mathrm {DPLL}\) to drunken \(\mathrm {DPLL}(\oplus )\).



The authors are grateful to Dmitry Sokolov for fruitful discussions.


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Authors and Affiliations

  1. 1.Steklov Institute of MathematicsSaint PetersburgRussia

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