Almost 2-SAT is Fixed-Parameter Tractable
Abstract
Description
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 $O(15^k*k*m^3)$ and thus we show that this problem is fixed-parameter tractable.
This new version fixes the bug found by Somnath Sikdar in the proof of Claim 8. In the repaired version the modification of the Almost 2-SAT problem called 2-SLASAT is no longer needed and only the modification called 2-ASLASAT remains relevant. Hence the whole manuscript is updated so that the 2-SLASAT problem is not mentioned there anymore
This new version fixes the bug found by Somnath Sikdar in the proof of Claim 8. In the repaired version the modification of the Almost 2-SAT problem called 2-SLASAT is no longer needed and only the modification called 2-ASLASAT remains relevant. Hence the whole manuscript is updated so that the 2-SLASAT problem is not mentioned there anymore