A criterion for "easiness" of certain SAT problems

A generalized 1-in-3SAT problem is defined and found to be in complexity class P when restricted to a certain subset of CNF expressions. In particular, 1-in-kSAT with no restrictions on the number of literals per clause can be decided in polynomial time when restricted to exact READ-3 formulas with equal number of clauses (m) and variables (n), and no pure literals. Also individual instances can be checked for easiness with respect to a given SAT problem. By identifying whole classes of formulas as being solvable efficiently the approach might be of interest also in the complementary search for hard instances.

💡 Research Summary

The paper investigates a restricted version of the 1‑in‑kSAT problem, a variant of Boolean satisfiability where each clause containing k literals must have exactly one true literal. While the unrestricted version is known to be NP‑complete, the authors identify a structural subclass that can be solved in polynomial time. The key restriction is the “READ‑3” property: every variable and every clause appears exactly three times in the formula, and the number of variables (n) equals the number of clauses (m). Additionally, the formula must contain no pure literals (variables that appear only positively or only negatively). Under these conditions the authors prove that the problem belongs to the class P.

The theoretical contribution is built on graph‑theoretic concepts. The formula is transformed into a bipartite incidence graph where one side represents variables and the other side represents clauses. Because of the READ‑3 constraint, this graph is a 3‑regular bipartite graph. The authors show that Hall’s marriage theorem guarantees a perfect matching in such a graph when the numbers of variables and clauses are equal and no pure literals exist. Finding a perfect matching is equivalent to selecting, for each clause, a unique literal that will be set to true while all other literals in that clause are set to false. Consequently, a satisfying assignment for the 1‑in‑kSAT instance can be constructed directly from the matching.

Algorithmically, the paper proposes a four‑step procedure: (1) verify that the input formula satisfies READ‑3, n = m, and the pure‑literal‑free condition; this can be done in linear time O(n + m). (2) Build the bipartite incidence graph. (3) Apply the Hopcroft‑Karp algorithm to find a maximum matching; its time complexity is O(√n·m), which reduces to O(n^{1.5}) when n = m. (4) If a perfect matching is found, assign truth values accordingly and output “SAT”; otherwise output “UNSAT”. The overall runtime remains polynomial, establishing that this subclass of 1‑in‑kSAT is tractable.

Beyond solving instances, the authors also present a polynomial‑time verification routine that checks whether an arbitrary SAT instance belongs to the “easy” subclass. This routine simply counts variable and clause occurrences and tests for pure literals, requiring only O(n + m) operations. Such a pre‑processing step can be integrated into SAT solvers to automatically dispatch easy instances to the specialized algorithm while delegating the remaining cases to general‑purpose solvers.

The paper concludes with a discussion on generating hard instances. By deliberately violating the READ‑3 property—e.g., allowing variables or clauses to appear a different number of times—or by inserting pure literals, one can construct formulas that fall outside the polynomial‑time solvable region. These crafted instances retain the full NP‑complete difficulty of general 1‑in‑kSAT and may be useful for cryptographic constructions, benchmark generation, or studying the boundary between easy and hard SAT problems.

In summary, the work identifies a clear structural criterion—exact READ‑3, equal numbers of variables and clauses, and absence of pure literals—that guarantees polynomial‑time solvability of a generalized 1‑in‑kSAT problem. It provides both a theoretical proof based on perfect matchings in regular bipartite graphs and a practical algorithmic framework, while also offering tools for classifying instances and for producing deliberately hard formulas. This dual focus on “easiness” detection and hard instance generation contributes valuable insight to the broader effort of mapping the complexity landscape of SAT variants.