On Parallel Partial Solutions and Approximation Schemesfor Local Consistency in Networks of Constraints

  • Authors:

  • N. D. Dendris;L. M. Kirousis;Y. C. Stamatiou;D. M. Thilikos

  • Affiliations:

  • Department of Computer Engineering and Informatics, Patras University, Rio, 265 00 Patras, Greece;Department of Computer Engineering and Informatics, Patras University, Rio, 265 00 Patras, Greece;Department of Computer Engineering and Informatics, Patras University, Rio, 265 00 Patras, Greece;Department of Computer Engineering and Informatics, Patras University, Rio, 265 00 Patras, Greece

  • Venue:
  • Constraints
  • Year:
  • 2000

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Abstract

A constraintnetwork is arc consistent if any value of any of its variablesis compatible with at least one value of any other variable.The Arc Consistency Problem (ACP) consists in filtering out valuesof the variables of a given network to obtain one that is arcconsistent, without eliminating any solution. ACP is known tobe inherently sequential, or P-complete, so in this paper weexamine some weaker versions of it and their parallel complexity.We propose several natural approximation schemes for ACP andshow that they are also P-complete. In an attempt to overcomethese negative results, we turn our attention to the problemof filtering out values from the variables so that each valuein the resulting network is compatible with at least one valueof not necessarily all, but a constant fraction of the othervariables. We call such a network partially arc consistent. Wegive a parallel algorithm that, for any constraint network, outputsa partially arc consistent subnetwork of it in sublinear ( O(\sqrt{n}\log{n})) parallel time using O(n^2) processors.This is the first (to our knowledge) sublinear-time parallelalgorithm with polynomially many processors that guarantees thatin the resulting network every value is compatible with at leastone value in at least a constant fraction of the remaining variables.Finally, we generalize the notion of partiality to the k-consistencyproblem.