Consistencies for Ultra-Weak Solutions in Minimax Weighted CSPs Using the Duality Principle

  • Arnaud Lallouet
  • Jimmy H. M. Lee
  • Terrence W. K. Mak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7514)


Minimax Weighted Constraint Satisfaction Problems (formerly called Quantified Weighted CSPs) are a framework for modeling soft constrained problems with adversarial conditions. In this paper, we describe novel definitions and implementations of node, arc and full directional arc consistency notions to help reduce search space on top of the basic tree search with alpha-beta pruning for solving ultra-weak solutions. In particular, these consistencies approximate the lower and upper bounds of the cost of a problem by exploiting the semantics of the quantifiers and reusing techniques from both Weighted and Quantified CSPs. Lower bound computation employs standard estimation of costs in the sub-problems used in alpha-beta search. In estimating upper bounds, we propose two approaches based on the Duality Principle: duality of quantifiers and duality of constraints. The first duality amounts to changing quantifiers from min to max , while the second duality re-uses the lower bound approximation functions on dual constraints to generate upper bounds. Experiments on three benchmarks comparing basic alpha-beta pruning and the six consistencies from the two dualities are performed to confirm the feasibility and efficiency of our proposal.


constraint optimization soft constraint satisfaction minimax game search consistency algorithms 


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Arnaud Lallouet
    • 1
  • Jimmy H. M. Lee
    • 2
  • Terrence W. K. Mak
    • 2
  1. 1.Université de Caen, GREYCCaen CedexFrance
  2. 2.The Chinese University of Hong KongShatinHong Kong

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