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Log-supermodular functions, functional clones and counting CSPs

Authors Andrei A. Bulatov, Martin Dyer, Leslie Ann Goldberg, Mark Jerrum

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Andrei A. Bulatov
Martin Dyer
Leslie Ann Goldberg
Mark Jerrum

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Andrei A. Bulatov, Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum. Log-supermodular functions, functional clones and counting CSPs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 302-313, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)


Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post's lattice. 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 our study, we assume that non-negative unary functions (weights) are available. Given this, we prove 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 as any problem in #P. Furthermore, any non-trivial functional clone (in a sense that will be made precise below) contains the binary function "implies". As a consequence, all non-trivial counting CSPs (with non-negative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately.
  • counting constraint satisfaction problems
  • approximation
  • complexity


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