Fractals for kernelization lower bounds, with an application to length-bounded cut problems

Bodlaender et al.'s [6] cross-composition technique is a popular method for excluding polynomialsize problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of cross-compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. Roughly speaking, our new technique combines the advantages of serial and parallel composition. In particular, answering an open question of Golovach and Thilikos [13], we show that, unless NP ⊆ coNP / poly, the NP-hard Length- Bounded Edge-Cut problem (delete at most k edges such that the resulting graph has no s-t path of length shorter than ℓ) parameterized by the combination of k and ℓ has no polynomialsize problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex deletion problems.