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Kernel bounds for disjoint cycles and disjoint paths

Abstract : In this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . Thus, these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand. We build upon recent results by Bodlaender et al. [6] and Fortnow and Santhanam [20], that show that NP-complete problems that are ‘or-compositional’ do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . To this machinery, we add a notion of transformation, and obtain that Disjoint Cycles, and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . For the proof, we introduce a problem on strings, called Disjoint Factors, and first show that this problem has no polynomial kernel unless N P ⊆ c o N P / p o l y . We also show that the related Disjoint Cycles Packing problem has a kernel of size O ( k log k ) .
Keywords : Lower bounds
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00806805
Contributor : Alexandre Pinlou <>
Submitted on : Tuesday, April 2, 2013 - 1:47:19 PM
Last modification on : Saturday, September 11, 2021 - 3:16:55 AM

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Hans L. Bodlaender, Stéphan Thomassé, Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theoretical Computer Science, Elsevier, 2011, 412, pp.4570-4578. ⟨10.1016/j.tcs.2011.04.039⟩. ⟨lirmm-00806805⟩

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