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 2SAT’, ’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 ∗ k ∗m) and thus we show that this problem is fixed-parameter