We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post's lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving reductions in the complexity class #RH\Pi_1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respect to approximation-preserving reductions, meaning that there is no fully polynomial randomised approximation scheme for counting satisfying assignments unless NP=RP.
Deep Dive into An approximation trichotomy for Boolean #CSP.
We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language specifying the relations that may be used in constraints. If every relation in the constraint language is affine then the number of satisfying assignments can be exactly counted in polynomial time. Otherwise, if every relation in the constraint language is in the co-clone IM_2 from Post’s lattice, then the problem of counting satisfying assignments is complete with respect to approximation-preserving reductions in the complexity class #RH\Pi_1. This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately counting the number of independent sets in a bipartite graph. For every other fixed constraint language, the problem is complete for #P with respec
This paper gives a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance. Such problems are parameterised by a constraint language Γ which specifies relations that may be used in constraints. In the Boolean case, the relations are on a domain which has two elements. Then #CSP(Γ) will denote the problem of determining the number of (distinct) satisfying assignments of a CSP instance with constraint language Γ. Further details are given in Section 1.1 below.
Creignou and Hermann [6] have given a dichotomy theorem for the exact counting problem. They have shown that if every relation in Γ is affine, then #CSP(Γ) is in FP. Otherwise, it is #P-complete. The complexity classes FP and #P are the analogues of P and NP for counting problems. FP is the class of functions computable in deterministic polynomial time. #P is the class of integer functions that can be expressed as the number of accepting computations of a polynomial-time non-deterministic Turing machine.
In this paper we build on previous work on the complexity of approximate counting to identify a trichotomy in the complexity of approximate counting for Boolean #CSP.
Together with Greenhill [9], we have previously studied approximationpreserving reductions (AP-reductions) between counting problems. We will give details of AP-reductions in Section 1.2. For now it suffices to note that if an AP-reduction exists from a counting problem f to a counting problem g and g has a Fully Polynomial Randomised Approximation Scheme (FPRAS) then f also has an FPRAS.
If an AP-reduction from f to g exists we write f ≤ AP g, and say that f is AP-reducible to g. If f ≤ AP g and g ≤ AP f then we say that f and g are AP-interreducible, and write f = AP g.
We previously identified [9] three natural classes of counting problems that are interreducible under AP-reductions. These are (i) those problems that have an FPRAS, (ii) those problems that are complete for #P with respect to AP-reducibility, and a third class of intermediate complexity. Two counting problems played a special role in [9].
Instance. A Boolean formula ϕ in conjunctive normal form.
Output. The number of satisfying assignments of ϕ.
Name. #BIS.
Output. The number of independent sets in B.
All problems in #P are AP-reducible to #SAT (see [9,Section 3]). Thus #SAT is complete for #P with respect to AP-reducibility. This means that #SAT cannot have an FPRAS unless NP = RP. The same is true of any problem in #P to which #SAT is AP-reducible.
We showed in [9,Sections 4,5] that #BIS is AP-interreducible with many other natural counting problems such as counting downsets in a partial order. Moreover, #BIS is complete for #RHΠ 1 , a logically-defined subclass of #P, with respect to AP-reductions.
The main theorem of our current paper (Theorem 3) shows that every problem #CSP(Γ) falls neatly into one of the three classes from [9]: If every relation in Γ is affine, then trivially #CSP(Γ) has an FPRAS since it is in FP. Otherwise, if every relation in Γ is in a certain set IM 2 , then #CSP(Γ) = AP #BIS. Otherwise #CSP(Γ) = AP #SAT. A formal definition of IM 2 appears in Section 1.4 -it is the set of relations which can be expressed as conjunctions involving only binary implication and unary relations.
It is worth pointing out that, while every problem #CSP(Γ) falls into one of the three approximation classes from [9], the three classes may well not provide a partition of all approximate counting problems in #P. For example, the problem of approximately counting 3-colourings of a bipartite graph is a problem that may well lie between #BIS and #SAT in approximability (see [9]).
Constraint Satisfaction, which originated in Artificial Intelligence, provides a general framework for modelling decision problems, and has many practical applications. (See, for example [18].) Decisions are modelled by variables, which are subject to constraints, modelling logical and resource restrictions. The paradigm is sufficiently broad that many interesting problems can be modelled, from satisfiability problems to scheduling problems and graph-theory problems. Understanding the complexity of constraint satisfaction problems has become a major and active area within computational complexity [7,11].
A Constraint Satisfaction Problem (CSP) typically has a finite domain, which we denote by {0, . . . , q -1} for a positive integer q. In this paper we are interested in the Boolean case q = 2. A constraint language Γ with domain {0, . . . , q -1} is a set of relations on {0, . . . , q -1}. For example, take q = 2. The relation R = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)} is a 3-ary relation on the domain {0, 1}, with four tuples.
Once we have fixed a constraint language Γ, an instance of the CSP is a set of variables V = {v 1 , . . . , v n } and a set of constraints. Each constraint has a scope, which is a tuple of variables (for example, (v 4 , v 5 , v 1 ))
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