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Author Bulatov, Andrei A. ♦ Dyer, Martin ♦ Goldberg, Leslie Ann ♦ Jerrum, Mark ♦ Mcquillan, Colin
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2013
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Approximation algorithms ♦ Computational complexity ♦ Constraint satisfaction problems ♦ Counting problems
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.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2013-10-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 60
Issue Number 5
Page Count 36
Starting Page 1
Ending Page 36

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Source: ACM Digital Library