Even faster algorithm for set splitting!

Abstract

In p-Set Splitting 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 kernelization and algorithmic view points. 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(1.9630k + 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, will turn out to be an important tool from combinatorial optimization in obtaining kernelization algorithms. © 2009 Springer-Verlag.