# Kernelization of Packing Problems

## TL;DR

This paper establishes tight lower bounds on kernel sizes for packing problems like d-Set Matching under complexity assumptions, and introduces new polynomial kernels for certain graph packing problems.

## Contribution

It provides the first tight lower bounds for kernel sizes of packing problems and develops a general scheme for proving such bounds.

## Key findings

- Tight lower bounds for d-Set Matching kernel sizes.
- Polynomial kernels with O(k^2.5) edges for vertex-disjoint paths of length three.
- A general reduction scheme for establishing kernel lower bounds.

## Abstract

Kernelization algorithms are polynomial-time reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, d-Set Matching for integers d>2 is the problem of finding a matching of size at least k in a given d-uniform hypergraph and has kernels with O(k^d) edges. Bodlaender et al. [JCSS 2009], Fortnow and Santhanam [JCSS 2011], Dell and Van Melkebeek [JACM 2014] developed a framework for proving lower bounds on the kernel size for certain problems, under the complexity-theoretic hypothesis that coNP is not contained in NP/poly. Under the same hypothesis, we show tight lower bounds for the kernelization of d-Set Matching and other packing problems. Our bounds are tight for d-Set Matching: It does not have kernels with O(k^{d-{\epsilon}}) edges for any {\epsilon}>0 unless the hypothesis fails. By reduction, this transfers to a bound of O(k^{d-1-{\epsilon}}) for the problem of finding k vertex-disjoint cliques of size d in standard graphs. Obtaining tight bounds for graph packing problems is challenging: We make first progress in this direction by showing non-trivial kernels with O(k^2.5) edges for the problem of finding k vertex-disjoint paths of three edges each. If the paths have d edges each, we improve the straightforward O(k^{d+1}) kernel to a uniform polynomial kernel where the exponent of the kernel size is independent of k. Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(k^{d-{\epsilon}}) for a problem in d-uniform hypergraphs, one should reduce from a carefully chosen d-partite problem that is still NP-hard. As an illustration, we apply this scheme to the vertex cover problem, which allows us to replace the number-theoretical construction by Dell and Van Melkebeek [JACM 2014] with shorter elementary arguments.