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(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 nding k vertex-disjoint cliques of size d in standard graphs. It is natur...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Holger Dell, Dániel Marx
• Keywords: Algorithms, Kernel Size, Polynomial Time, SODA 2012, Tight Bounds