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    The expressibility of functions on the Boolean domain, with applications to Counting CSPs 
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    The expressibility of functions on the Boolean domain, with applications to Counting CSPs

    Volume
    60
    Pagination
    ? - ? (36)
    Publisher
    ACM Digital Library
    Publisher URL
    http://dl.acm.org/citation.cfm?id=2528401&CFID=463360810&CFTOKEN=40583309 http://dl.acm.org/citation.cfm?id=2528401&CFID=463360810&CFTOKEN=40583309
    DOI
    10.1145/2528401
    Issue
    5
    ISSN
    0004-5411
    Metadata
    Show full item record
    Abstract
    An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function “implies”. As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the “implies” clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available
    Authors
    Bulatov, A; Dyer, M; Goldberg, LA; Jerrum, M; McQuillan, C
    URI
    http://qmro.qmul.ac.uk/xmlui/handle/123456789/10295
    Collections
    • Mathematics [1001]
    Language
    English
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