cs.CC — Computational Complexity
A Sublinear Treewidth Exponent for Binary Constraint Satisfaction
We give a deterministic algorithm for binary constraint satisfaction whose exponent in the input size is sublinear in treewidth on a class of graphs with unbounded treewidth. There are a class G of graphs of bounded maximum degree, a constant c, and a function f such that one algorithm correctly decides every binary CSP instance and, on instances I with primal graph G ∈ G, runs in time f(G) · ||I||^(c · tw(G) / sqrt(log tw(G))) . This contradicts the literal formulation of Conjecture 1.4 of Marx (2010). The algorithm follows from a domain-uniform counting bound: for every fixed maximum degree ∆, complete-table in- stances with k variables and domain size D ≥ 2 can be counted in time LO∆ (1)DO∆ (k/√log k) for sufficiently large k. The constants are independent of D and of the constraint relations. We obtain this bound from the tensor-network recursion of Claude (Anthropic), and use attributed graph foundations from OpenAI to construct the required expander family. Support pruning connects the counting routine to sparse tuple-list inputs without expanding the original domain. The algo- rithm, its native finite-alphabet machine implementation, and the conjecture counterexample are formalized in Lean 4.