Faster Parameterized Algorithms Using Linear Programming

We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O*(2.618^k) algorithm for the problem. Here, k is the excess of the vertex cover size over the LP optimum, and we write O*(f(k)) for a time complexity of the form O(f(k)n^O(1)). We proceed to show that a more sophisticated branching algorithm achieves a running time of O*(2.3146^k).

Following this, using previously known as well as new reductions, we give O*(2.3146^k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, and Almost 2-SAT, and O*(1.5214^k) algorithms for König Vertex Deletion and Vertex Cover parameterized by the size of the smallest odd cycle transversal and König vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for Odd Cycle Transversal—this is the first algorithm that improves on the dependence on k of the seminal O*(3^k) algorithm of Reed, Smith, and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k − clog k vertices. Our kernel is simpler than previously known kernels achieving the same size bound.