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⊆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⊆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⊆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⊆coNP/poly. We also show that the related Disjoint Cycles Packing problem has a kernel of size O(klogk).