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    Soft constraints of difference and equality

    Hebrard, E. and Marx, D. and O'Sullivan, B. and Razgon, Igor (2011) Soft constraints of difference and equality. Journal of Artificial Intelligence Research 41 , pp. 97-130. ISSN 1076-9757.

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    Abstract

    In many combinatorial problems one may need to model the diversity or similarity of assignments in a solution. For example, one may wish to maximise or minimise the number of distinct values in a solution. To formulate problems of this type, we can use soft variants of the well known AllDifferent and AllEqual constraints. We present a taxonomy of six soft global constraints, generated by combining the two latter ones and the two standard cost functions, which are either maximised or minimised. We characterise the complexity of achieving arc and bounds consistency on these constraints, resolving those cases for which NP-hardness was neither proven nor disproven. In particular, we explore in depth the constraint ensuring that at least k pairs of variables have a common value. We show that achieving arc consistency is NP-hard, however achieving bounds consistency can be done in polynomial time through dynamic programming. Moreover, we show that the maximum number of pairs of equal variables can be approximated by a factor 1/2 with a linear time greedy algorithm. Finally, we provide a fixed parameter tractable algorithm with respect to the number of values appearing in more than two distinct domains. Interestingly, this taxonomy shows that enforcing equality is harder than enforcing difference.

    Metadata

    Item Type: Article
    School: Birkbeck Schools and Departments > School of Business, Economics & Informatics > Computer Science and Information Systems
    Depositing User: Sarah Hall
    Date Deposited: 01 Aug 2013 15:18
    Last Modified: 01 Aug 2013 15:18
    URI: http://eprints.bbk.ac.uk/id/eprint/7917

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