David Richerby

An Effective Dichotomy for the Counting Constraint Satisfaction Problem

Bulatov [Proceedings of the

The complexity of weighted and unweighted #CSP

35

The complexity of approximating bounded-degree Boolean #CSP

th International Colloquium on Automata, Languages and Programming (Part 1), Lecture Notes in Comput. Sci. 5125, Springer, New York, 2008, pp. 646--661] gave a dichotomy for the counting constraint satisfaction problem \#CSP. A problem from \#CSP is characterized by a constraint language

The #CSP Dichotomy is Decidable

\Gamma\!

Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs

, a fixed, finite set of relations over a finite domain

On the fixation probability of superstars

D

Graph Searching in a Crime Wave

. An instance of the problem uses these relations to constrain an arbitrarily large finite set of variables. Bulatov showed that the problem of counting the satisfying assignments of instances of any problem from \#CSP is either in polynomial time (FP) or is \#P-complete. His proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field. We give an elementary proof of Bulatovs dichotomy, based on succinct representations, which we call frames, of a class of highly structured relations, which we call strongly rectangular. We show that these are precisely the relation...

Absorption Time of the Moran Process

We give some reductions among problems in (nonnegative) weighted #CSP which restrict the class of functions that needs to be considered in computational complexity studies. Our reductions can be applied to both exact and approximate computation. In particular, we show that the recent dichotomy for unweighted #CSP can be extended to rational-weighted #CSP.

The power of counting logics on restricted classes of finite structures

The degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

Fixed-point Logics with Nondeterministic Choice

Bulatov (2008) and Dyer and Richerby (2010) have established the following dichotomy for the counting constraint satisfaction problem (#CSP): for any constraint language Gamma, the problem of computing the number of satisfying assignments to constraints drawn from Gamma is either in FP or is #P-complete, depending on the structure of Gamma. The principal question left open by this research was whether the criterion of the dichotomy is decidable. We show that it is; in fact, it is in NP.