Kernelization of packing problems

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 ≥ 3 is the problem of nding a matching of size at least k in a given d-uniform hypergraph and has kernels with O(k) edges. Recently, Bodlaender et al. [ICALP 2008], Fortnow and Santhanam [STOC 2008], Dell and Van Melkebeek [STOC 2010] 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 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(kd− ) edges for any > 0 unless the hypothesis fails. By reduction, this transfers to a bound of O(kd−1− ) for the problem of nding k vertex-disjoint cliques of size d in standard graphs. It is natural to ask for tight bounds on the kernel sizes of such graph packing problems. We make rst progress in that direction by showing nontrivial kernels with O(k) edges for the problem of nding k vertex-disjoint paths of three edges each. This does not quite match the best lower bound of O(k2− ) that we can prove. Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(kd− ) 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 [STOC 2010] with shorter elementary arguments.