The Complexity of Approximating Bounded-Degree Boolean sharp CSP

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.

💡 Research Summary

The paper investigates the approximation complexity of counting solutions to Boolean constraint satisfaction problems (CSPs) when the instance is bounded‑degree, i.e., each variable appears in at most Δ constraints. The authors focus on languages that contain the two constant unary relations {0} and {1}. Their main contribution is a complete trichotomy for Δ ≥ 6 and a nuanced classification for smaller Δ.

1. Affine languages – If every relation in the constraint language is affine (equivalently, each constraint can be expressed as a system of linear equations over GF(2)), the counting problem is solvable exactly in polynomial time, regardless of the degree bound. The algorithm relies on Gaussian elimination to compute the dimension of the solution space.

2. Implication‑only languages – When each relation can be written as a conjunction of the constants {0}, {1} and the binary implication (x → y), the problem is AP‑equivalent to #BIS, the problem of approximately counting independent sets in bipartite graphs. The authors construct a polynomial‑time AP‑reduction from the bounded‑degree CSP to #BIS by translating each implication constraint into an edge of a bipartite graph, and they also give the reverse reduction, establishing exact equivalence. Consequently, the problem inherits the conjectured intermediate complexity of #BIS: it is believed to be neither in P nor #P‑complete, and no FPRAS is known.

3. All other languages – For any Boolean language that is neither affine nor implication‑only, the authors prove that no fully polynomial‑randomized approximation scheme (FPRAS) exists unless NP = RP. This hardness is shown by an AP‑reduction from #SAT (or #3‑SAT) to the bounded‑degree CSP, demonstrating that an FPRAS would collapse the randomized polynomial hierarchy.

When the degree bound Δ is below 6, the landscape becomes richer. The paper shows that for Δ = 3, 4, 5 the counting problem can be related to the problem of approximately counting independent sets in uniform hypergraphs (3‑uniform, 4‑uniform, 5‑uniform respectively). These hypergraph independent‑set problems are themselves of unknown exact complexity and are believed to be harder than #BIS but easier than full #P‑hardness. Thus, new intermediate regimes appear that are not captured by the Δ ≥ 6 trichotomy.

Methodologically, the work combines linear‑algebraic techniques for affine cases, graph‑theoretic constructions for implication‑only cases, and standard AP‑reduction machinery for hardness proofs. The authors also develop novel reductions that respect the degree bound, ensuring that the transformed instances do not exceed the prescribed Δ.

The significance of the results lies in extending the well‑known dichotomy/trichotomy theorems for Boolean #CSPs to the realistic setting where each variable participates in only a few constraints. The threshold Δ = 6 emerges as a natural boundary: above it the classic language‑based classification suffices, while below it the structure of the underlying hypergraph becomes decisive. This work therefore bridges the gap between algebraic classifications of CSPs and combinatorial properties of sparse instances, opening avenues for future research on the exact complexity of hypergraph independent‑set counting and on designing approximation algorithms for low‑degree CSPs.