Σ-All Different: Softening AllDifferent in Weighted CSPs

A soft version of the well-known global constraint AllDifferent has been recently introduced for the Max-CSP framework ([9, 15, 16]). In this paper, we propose to soften AllDifferent in the Weighted CSP framework that is more general. We extend the two semantics of violation proposed in [9]: the first one is based on variables and the second one on the decomposition into a set of binary constraints of difference. For the first semantic, we propose a polynomial algorithm which maintains hyper- arc consistency. For the second one, we prove that checking hyper-arc consistency is an NP-Hard problem. So, we propose to maintain a local consistency using a filtering based on lower bounds computation. Finally, we present some experimental results and draw a few perspectives.

[1]  Gilles Pesant,et al.  On global warming: Flow-based soft global constraints , 2006, J. Heuristics.

[2]  P. Hall.  On Representatives of Subsets , 1935 .

[3]  D. R. Fulkerson,et al.  Constructing Maximal Dynamic Flows from Static Flows , 1958 .

[4]  Willem Jan van Hoeve,et al.  A Hyper-arc Consistency Algorithm for the Soft Alldifferent Constraint , 2004, CP.

[5]  Jean-Charles Régin,et al.  A Filtering Algorithm for Constraints of Difference in CSPs , 1994, AAAI.

[6]  Thomas Schiex,et al.  On the Complexity of Compact Coalitional Games , 2009, IJCAI.

[7]  Christian Bessiere,et al.  A General Conflict-Set Based Framework for Partial Constraint Satisfaction , 2008 .

[8]  Richard J. Wallace,et al.  Partial Constraint Satisfaction , 1989, IJCAI.

[9]  Patrice Boizumault,et al.  Combining VNS with constraint programming for solving anytime optimization problems , 2008, Eur. J. Oper. Res..

[10]  Christian Bessiere,et al.  Specific Filtering Algorithms for Over-Constrained Problems , 2001, CP.

[11]  Philippe Refalo,et al.  Impact-Based Search Strategies for Constraint Programming , 2004, CP.

[12]  Simon de Givry,et al.  Radio Link Frequency Assignment , 1999, Constraints.

[13]  Yehuda Naveh,et al.  Generalizing AllDifferent: The SomeDifferent Constraint , 2006, CP.

[14]  Thomas Schiex,et al.  Semiring-Based CSPs and Valued CSPs: Frameworks, Properties, and Comparison , 1999, Constraints.

[15]  Jean-Charles Régin,et al.  Generalized Arc Consistency for Global Cardinality Constraint , 1996, AAAI/IAAI, Vol. 1.

[16]  Christian Bessiere,et al.  An Original Constraint Based Approach for Solving over Constrained Problems , 2000, CP.

[17]  Hadrien Cambazard,et al.  Identifying and Exploiting Problem Structures Using Explanation-based Constraint Programming , 2005, Constraints.