Kernelization of Packing Problems

Summary

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 finding a matching of size at least k in a given d-uniform hypergraph and has kernels with O(kd) 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 finding 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 first progress in that direction by showing non-trivial kernels with O(k2.5) edges for the problem of finding 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.