Weak compositions and their applications to polynomial lower bounds for kernelization

In this paper we use the notion of weak compositions to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1}* x N be two parameterized problems where the unparameterized version of L1 is NP-hard. Assuming coNP n NP/poly, our framework essentially states that composing t L1-instances each with parameter k, to an L2-instance with parameter k' ≤ t1/dkO(1), implies that L2 does not have a kernel of size O(kd−e) for any e > 0. We show two examples of weak composition and derive polynomial kernelization lower bounds for d-Bipartite Regular Perfect Code and d-Dimensional Matching, parameterized by the solution size k. By reduction, using linear parameter transformations, we then derive the following lower-bounds for kernel sizes when the parameter is the solution size k (assuming coNP n NP/poly): • d-Set Packing, d-Set Cover, d-Exact Set Cover, Hitting Set with d-Bounded Occurrences, and Exact Hitting Set with d-Bounded Occurrences have no kernels of size O(kd−3−e) for any e > 0. • Kd Packing and Induced K1,d Packing have no kernels of size O(kd−4−e) for any e > 0. • d-Red-Blue Dominating Set and d-Steiner Tree have no kernels of sizes O(kd−3−e) and O(kd−4−e), respectively, for any e > 0. Our results give a negative answer to an open question raised by Dom, Lokshtanov, and Saurabh [ICALP2009] regarding the existence of uniform polynomial kernels for the problems above. All our lower bounds transfer automatically to compression lower bounds, a notion defined by Harnik and Naor [SICOMP2010] to study the compressibility of NP instances with cryptographic applications. We believe weak composition can be used to obtain polynomial kernelization lower bounds for other interesting parameterized problems. In the last part of the paper we strengthen previously known super-polynomial kernelization lower bounds to super-quasi-polynomial lower bounds, by showing that quasi-polynomial kernels for compositional NP-hard parameterized problems implies the collapse of the exponential hierarchy. These bounds hold even the kernelization algorithms are allowed to run in quasipolynomial time.