The Complexity of Approximating Bounded-Degree Boolean #CSP (Extended Abstract)

The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum degree is at least 25 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 in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

💡 Research Summary

The paper investigates the approximate counting complexity of Boolean constraint satisfaction problems (#CSP) when the instances are bounded‑degree, i.e., each variable appears in at most Δ constraints. The authors assume that the constraint language Γ always contains the two unary constant relations {0} and {1}. Their main contribution is a complete trichotomy for the case Δ ≥ 25 and a refined classification for smaller degree bounds, linking several previously separate complexity regimes.

High‑degree regime (Δ ≥ 25).
Three mutually exclusive cases are identified:

1. Affine languages. If every relation in Γ is affine (i.e., each relation can be expressed as the solution set of a system of linear equations over GF(2)), the counting problem is solvable exactly in polynomial time. The authors show that the bounded‑degree restriction does not affect the classic Gaussian‑elimination algorithm; the number of satisfying assignments can be computed by solving a linear system and raising 2 to the dimension of the solution space.

2. Implication‑generated languages. If each relation in Γ can be expressed as a conjunction of the constants {0}, {1} and the binary implication relation (x → y), then #CSP(Γ) is AP‑equivalent to #BIS, the problem of approximately counting independent sets in bipartite graphs. The reduction builds a bipartite incidence graph where variables correspond to one side and constraints to the other; a satisfying assignment corresponds precisely to an independent set. Since #BIS is believed to be neither in FP nor #P‑complete (it occupies an “intermediate” status), no fully polynomial‑randomized approximation scheme (FPRAS) exists for these languages unless NP = RP.

3. All other languages. For any Γ that does not fall into the previous two categories, the authors prove AP‑hardness for #CSP(Γ) by constructing degree‑preserving gadgets that simulate arbitrary Boolean formulas while keeping the maximum variable degree at most 25. Consequently, under the standard assumption NP ≠ RP, no FPRAS exists for these languages.

Low‑degree regime (Δ < 25).
When the degree bound is smaller, additional complexity classes appear. The paper shows that for certain Γ containing implication together with negation, the counting problem becomes equivalent to counting independent sets in k‑uniform hypergraphs (#HyperIndSet). The difficulty then depends on both the degree Δ and the hyperedge size k:

• For Δ = 3 and k = 2 (i.e., ordinary graphs of degree three) the problem is again #BIS‑hard.
• For Δ = 4 and k = 3 the status remains open, but known techniques do not yield an FPRAS.
• For larger Δ (≥ 5) but with bounded hyperedge size, the authors identify subclasses where an FPRAS is possible, typically when the hypergraph structure is “linear” (edges intersect in at most one vertex).

Technical tools.
The classification relies on a combination of AP‑reductions, holographic reductions, and a novel “degree‑preserving gadget” that allows variable cloning without exceeding the degree bound. The gadget is carefully designed to enforce logical equivalence while using at most 24 auxiliary constraints per variable, which is why the threshold 25 emerges naturally in the analysis. The paper also employs pinning (forcing variables to constants) to handle the mandatory presence of {0} and {1} in Γ.

Implications and open problems.
The work extends the classic dichotomy for unrestricted Boolean #CSP (by Creignou, Hermann, and others) to the bounded‑degree setting, revealing that degree constraints can both simplify (affine case) and complicate (implication‑generated case) the landscape. The authors leave several directions for future research: lowering the constant 25, tightening the boundary between #BIS‑hard and tractable cases for low degrees, and deepening the understanding of the intermediate complexity class represented by #HyperIndSet. Overall, the paper provides a comprehensive map of the approximability of bounded‑degree Boolean #CSPs, unifying several strands of counting complexity under a single framework.