Abstract
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(1.9630k + 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.
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Lokshtanov, D., Saurabh, S. (2009). Even Faster Algorithm for Set Splitting!. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_24
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DOI: https://doi.org/10.1007/978-3-642-11269-0_24
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