A Probabilistic Algorithm for k-SAT and Constraint Satisfaction Problems

  • Authors:

  • Uwe Schöning

  • Affiliations:

  • -

  • Venue:
  • FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
  • Year:
  • 1999

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Abstract

We present a simple probabilistic algorithm for solving k-SAT, and more generally, for solving constraint satisfaction problems (CSP). The algorithm follows a simple local-search paradigm: randomly guess an initial assignment and then, guided by those clauses (constraints) that are not satisfied, by successively choosing a random literal from such a clause and flipping the corresponding bit, try to find a satisfying assignment. If no satisfying assignment is found after O(n) steps, start over again. Our analysis shows that for any satisfiable k-CNF formula with n variables this process has to be repeated only t times, on the average, to find a satisfying assignment, where t is within a polynomial factor of (2(1-1/k))n. This is the fastest (and also the simplest) algorithm for 3-SAT known up to date. We consider also the more general case of a CSP with n variables, each variable taking at most d values, and constraints of order l, and analyze the complexity of the corresponding (generalized) algorithm. It turns out that any CSP can be solved with complexity about (d(1-1/l))n.