A Linear Vertex Kernel for Maximum Internal Spanning Tree
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
New lower bound on max cut of hypergraphs with an application to r-set splitting
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Discrete Optimization
A linear vertex kernel for maximum internal spanning tree
Journal of Computer and System Sciences
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In p -Set Splitting we are given a universe U, a family $\cal F$ 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 $\cal F$ 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 $(U,{\cal F},k)$ 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 k + 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.