LP can be a cure for Parameterized Problems

We investigate the parameterized complexity of Vertex Cover parameterized above the optimum value of the linear programming (LP) relaxation of the integer linear programming formulation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that even the most straightforward branching algorithm (after some preprocessing) results in an O∗(2.6181r) algorithm for the problem where r is the excess of the vertex cover size over the LP optimum. We write O∗(f(k)) for a time complexity of the form O(f(k)nO(1)), where f(k) grows exponentially with k. Then, using known and new reductions, we give O∗(2.6181k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an O∗(1.6181k) algorithm for Kon̈ig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The notable improvement is the bound for Odd Cycle Transversal for which this is the first major improvement after the first algorithm that showed it fixed-parameter tractable in 2003. We also observe that using our algorithm, one can obtain a simple kernel for the classical vertex cover problem with at most 2k −O(log k) vertices. 1998 ACM Subject Classification G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems