The complexity of approximating conservative counting CSPs
We study the complexity of approximation for a weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known for the Boolean domain. We give a classification for problems with general finite domain. We define weak log-modularity and weak log-supermodularity, and show that #CSP(F) is in FP if F is weakly log-modular. Otherwise, it is at least as hard to approximate as #BIS, counting independent sets in bipartite graphs, which is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, we show that #CSP(F) is as easy as Boolean log-supermodular weighted #CSP. Otherwise, it is NP-hard to approximate. Finally, we give a trichotomy for the arity-2 case.
Then, #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT, the problem of approximately counting satisfying assignments of a CNF Boolean formula.
counting constraint satisfaction problem
approximation
complexity
148-159
Regular Paper
Xi
Chen
Xi Chen
Martin
Dyer
Martin Dyer
Leslie Ann
Goldberg
Leslie Ann Goldberg
Mark
Jerrum
Mark Jerrum
Pinyan
Lu
Pinyan Lu
Colin
McQuillan
Colin McQuillan
David
Richerby
David Richerby
10.4230/LIPIcs.STACS.2013.148
Creative Commons Attribution-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nd/3.0/legalcode