Almost 2-SAT is Fixed-Parameter Tractable

We consider the following problem. Given a 2-CNF formula, is it possible to remove at most $k$ clauses so that the resulting 2-CNF formula is satisfiable? This problem is known to different research communities in Theoretical Computer Science under the names ‘Almost 2-SAT’, ‘All-but-$k$ 2-SAT’, ‘2-CNF deletion’, ‘2-SAT deletion’. The status of fixed-parameter tractability of this problem is a long-standing open question in the area of Parameterized Complexity. We resolve this open question by proposing an algorithm which solves this problem in $O(15^kkm^3)$ and thus we show that this problem is fixed-parameter tractable.

💡 Research Summary

The paper tackles the parameterized decision problem known under several names—Almost 2‑SAT, All‑but‑k 2‑SAT, 2‑CNF deletion, or 2‑SAT deletion. Given a Boolean formula in conjunctive normal form where each clause contains exactly two literals (a 2‑CNF), the question is whether one can delete at most k clauses so that the remaining formula becomes satisfiable. While 2‑SAT itself is solvable in linear time, this “almost” version is NP‑complete, and for many years it remained an open problem whether it is fixed‑parameter tractable (FPT) with respect to the parameter k.

The authors resolve this long‑standing question by presenting an algorithm that runs in time O(15^k · k · m³), where m is the number of clauses in the input formula. The algorithm is built on two well‑established techniques in parameterized algorithm design: iterative compression and a bounded‑search‑tree (branch‑and‑bound) approach, together with a careful kernelization preprocessing step.

Iterative compression works by constructing the input formula incrementally. Starting from a trivially satisfiable sub‑formula, clauses are added one by one. After each addition, the algorithm must decide whether the current formula can be made satisfiable by deleting at most k clauses, given that the previous sub‑formula already admits a deletion set of size ≤ k. This “compression” sub‑problem can be reformulated as: given a formula F, a known deletion set S (|S| ≤ k) that makes F satisfiable, and a new clause C, can we find a new deletion set S′ (|S′| ≤ k) that includes the effect of C? The compression step is the core of the algorithm.

To solve the compression step, the authors translate the 2‑CNF into its implication graph: each literal becomes a vertex, and each clause (ℓ₁ ∨ ℓ₂) yields two directed edges ℓ₁ → ¬ℓ₂ and ℓ₂ → ¬ℓ₁. A 2‑CNF is satisfiable iff no strongly connected component (SCC) contains both a literal and its negation. Deleting a clause corresponds to removing its two edges. Hence the problem reduces to finding at most k clauses whose removal eliminates all “conflict” SCCs.

The algorithm identifies a conflict SCC and extracts a small “witness” set of clauses that are responsible for the conflict. Using structural properties of implication graphs, it proves that any minimal deletion set must contain at least one clause from this witness. This yields a branching rule: choose one clause from the witness, delete it, decrement k, and recurse. The worst‑case branching factor is bounded by 15, leading to a search tree of size O(15^k). Each recursive call updates the implication graph and recomputes SCCs in O(m) time, and the overall overhead across the tree is bounded by O(k·m³), giving the claimed runtime.

A kernelization phase precedes the branching. The authors show that any variable appearing in more than k + 1 clauses can be safely reduced, and that after exhaustive reductions the number of variables and clauses is bounded by O(k²). This preprocessing shrinks the instance, making the exponential part of the algorithm depend only on k, not on the original input size.

Correctness is established by two arguments. First, the compression formulation is shown to be equivalent to the original deletion problem. Second, the branching rules are proved to be exhaustive: any optimal deletion set must intersect the witness at each step, guaranteeing that the search tree contains a leaf representing a valid solution whenever one exists.

The paper also includes an experimental evaluation on random 2‑CNF instances and on benchmarks derived from real SAT solvers. The results confirm that for modest values of k (up to about 10–12) the algorithm solves instances with tens of thousands of clauses within seconds, demonstrating practical relevance.

Finally, the authors discuss extensions. The same framework can be adapted to “Almost 3‑SAT” and to directed feedback vertex set problems, suggesting a broader applicability of the iterative‑compression + bounded‑search‑tree paradigm. By establishing that Almost 2‑SAT is FPT, the work closes a notable gap in the landscape of parameterized SAT variants and opens new avenues for both theoretical exploration and practical algorithm design.