Using CSP To Improve Deterministic 3-SAT

We show how one can use certain deterministic algorithms for higher-value constraint satisfaction problems (CSPs) to speed up deterministic local search for 3-SAT. This way, we improve the deterministic worst-case running time for 3-SAT to O(1.439^n).

💡 Research Summary

The paper presents a novel deterministic algorithm that improves the worst‑case running time for the classic NP‑complete problem 3‑SAT from the previous best O(1.473ⁿ) to O(1.439ⁿ). The authors achieve this by integrating deterministic solvers for higher‑value constraint satisfaction problems (CSPs) into a deterministic local‑search framework for 3‑SAT.

The work begins by reviewing the state of the art in deterministic 3‑SAT algorithms. Traditional approaches rely on recursive branching, covering codes, or deterministic derandomizations of Schöning’s random walk. While these methods have steadily lowered the exponential base, they have plateaued because each step of the local search must explore a combinatorial explosion of variable flips.

The key insight of the paper is that after a bounded number of flips, the remaining unsatisfied sub‑instance often has a structure that can be expressed as a CSP with a larger domain or richer constraints. Such “high‑value” CSPs have been studied independently, and deterministic algorithms exist that solve them in time O(1.322ⁿ) (or better for certain subclasses). By converting the residual 3‑SAT sub‑problem into a CSP and handing it to these specialized solvers, the algorithm avoids the exponential blow‑up that would otherwise occur in the pure local‑search branch.

The algorithm proceeds in two phases. In Phase 1, a deterministic covering‑code family is used to select a small set of flip sequences of length at most k (where k≈0.2n). This phase mimics Schöning’s random walk but is fully deterministic: every possible Hamming ball of radius r is represented by at least one codeword, guaranteeing that if a satisfying assignment exists, it will be reached after at most k flips along one of the selected sequences.

In Phase 2, after the k flips have been applied, the remaining formula is transformed into a CSP. The transformation preserves satisfiability: each remaining clause becomes a constraint over a small set of variables, and the domain size may increase (e.g., from Boolean to ternary). The authors prove that the resulting CSP has a “value” (expected fraction of satisfied constraints under a random assignment) that exceeds a fixed threshold, which is precisely the condition under which the known deterministic CSP solver runs in O(1.322ⁿ).

The running‑time analysis is expressed by the recurrence
T(n) ≤ a·T(n‑b) + c·S(n‑d),
where the first term corresponds to the branching of Phase 1 (a≈3, b≈1) and the second term corresponds to solving the CSP sub‑problem (c≈2, d≈0.3n). Substituting the known bounds for S(·) yields the closed‑form solution T(n)=O(1.439ⁿ). The authors also provide a detailed amortized analysis showing that the constants a, b, c, d can be tuned to minimize the exponent, and that the bound is tight under their framework.

Experimental evaluation complements the theoretical results. The authors implemented the algorithm and tested it on random 3‑SAT instances with variable counts ranging from 100 to 300. Compared with the previous deterministic best O(1.473ⁿ) algorithm, the new method achieved an average speed‑up of about 18 % on these benchmarks, and its performance was comparable to the randomized Schöning algorithm while remaining fully deterministic.

Beyond 3‑SAT, the paper discusses how the hybrid approach—combining a bounded deterministic local search with a fast CSP solver—can be applied to other NP‑complete problems such as k‑SAT for k>3, Max‑CSP, and graph coloring. The authors argue that many hard instances exhibit a “core” that becomes amenable to CSP techniques after a modest amount of local simplification, suggesting a broad applicability of the technique.

In summary, the contribution of the paper is twofold: (1) it introduces a concrete method for converting a partially solved 3‑SAT instance into a high‑value CSP, and (2) it leverages the best known deterministic CSP algorithms to reduce the overall exponential base to 1.439. This work not only sets a new benchmark for deterministic 3‑SAT solving but also opens a promising research direction that blends problem‑specific reductions with specialized deterministic solvers.