Kernel bounds for disjoint cycles and disjoint paths
In this paper, we show that the problems DISJOINT CYCLES and DISJOINT PATHS do not have polynomial kernels, unless NP subset of coNP/poly. 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 NP subset of coNP/poly. To this machinery, we add a notion of transformation, and obtain that DISJOINT CYCLES, and DISJOINT PATHS do not have polynomial kernels, unless NP subset of coNP/poly. For the proof, we introduce a problem on strings, called DISJOINT FACTORS, and first show that this problem has no polynomial kernel unless NP subset of coNP/poly. We also show that the related DISJOINT CYCLES PACKING problem has a kernel of size O(k log k). (C) 2011 Elsevier B.V. All rights reserved.
kernelization disjoint cycles data reduction fixed parameter complexity algorithms lower bounds