Even Faster Algorithm for Set Splitting!

In the p-Set Splitting problem we are given a universe U , a family F of subsets of U and a positive integer k and the objective is to find a partition of U into W and B such that there are at least k sets in F that have non-empty intersection with both B and W . In this paper we study p-Set Splitting from the view point of kernelization and parameterized algorithms. Given an instance (U,F , k) of p-Set Splitting, our kernelization algorithm obtains an equivalent instance with at most 2k sets and k elements in polynomial time. Finally, we give a fixed parameter tractable algorithm for p-Set Splitting running in time O(1.9630 +N), where N is the size of the instance. Both our kernel and our algorithm improve over the best previously known results. Our kernelization algorithm utilizes a classical duality theorem for a connectivity notion in hypergraphs. We believe that the duality theorem we make use of could become an important tool in obtaining kernelization algorithms.