A Discrete Lagrangian-Based Global-SearchMethod for Solving Satisfiability Problems

  • Authors:

  • Yi Shang;Benjamin W. Wah

  • Affiliations:

  • Department of Computer Engineering and Computer Science, University of Missouri, Columbia, MO 65211, USA. email: yshang@cecs.missouri.edu;Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, Urbana, IL 61801, USA. email: b-wah@uiuc.edu

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 1998

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Abstract

Satisfiability is a class of NP-complete problems that model a wide rangeof real-world applications. These problems are difficult to solve becausethey have many local minima in their search space, often trapping greedysearch methods that utilize some form of descent. In this paper, we proposea new discrete Lagrange-multiplier-based global-search method (DLM) forsolving satisfiability problems. We derive new approaches for applyingLagrangian methods in discrete space, we show that an equilibrium is reachedwhen a feasible assignment to the original problem is found and presentheuristic algorithms to look for equilibrium points. Our method and analysisprovides a theoretical foundation and generalization of local search schemesthat optimize the objective alone and penalty-based schemes that optimizethe constraints alone. In contrast to local search methods that restart froma new starting point when a search reaches a local trap, the Lagrangemultipliers in DLM provide a force to lead the search out of a local minimumand move it in the direction provided by the Lagrange multipliers. Incontrast to penalty-based schemes that rely only on the weights of violatedconstraints to escape from local minima, DLM also uses the value of anobjective function (in this case the number of violated constraints) toprovide further guidance. The dynamic shift in emphasis between theobjective and the constraints, depending on their relative values, is thekey of Lagrangian methods. One of the major advantages of DLM is that it hasvery few algorithmic parameters to be tuned by users. Besides the searchprocedure can be made deterministic and the results reproducible. Wedemonstrate our method by applying it to solve an extensive set of benchmarkproblems archived in DIMACS of Rutgers University. DLM often performs betterthan the best existing methods and can achieve an order-of-magnitudespeed-up for some problems.