We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language Gamma and a degree bound Delta, we study the complexity of #CSP_Delta(Gamma), which is the problem of counting satisfying assignments to CSP instances with constraints from Gamma and whose variables can appear at most Delta times. Our main result shows that: (i) if every function in Gamma is affine, then #CSP_Delta(Gamma) is in FP for all Delta, (ii) otherwise, if every function in Gamma is in a class called IM_2, then for all sufficiently large Delta, #CSP_Delta(Gamma) is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large Delta, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP_Delta(Gamma), even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.
@InProceedings{galanis_et_al:LIPIcs.ICALP.2017.27, author = {Galanis, Andreas and Goldberg, Leslie Ann and Yang, Kuan}, title = {{Approximating Partition Functions of Bounded-Degree Boolean Counting Constraint Satisfaction Problems}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.27}, URN = {urn:nbn:de:0030-drops-74099}, doi = {10.4230/LIPIcs.ICALP.2017.27}, annote = {Keywords: Constraint Satisfaction, Approximate Counting} }
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