Kernelization of packing problems
Holger Dell,Dániel Marx +1 more
- 17 Jan 2012
- pp 68-81
TL;DR: This work shows lower bounds for the kernelization of d-Set Matching and other packing problems, and applies this scheme to the vertex cover problem, which allows us to replace the number-theoretical construction by Dell and Van Melkebeek with shorter elementary arguments.
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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 ≥ 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−e) edges for any e > 0 unless the hypothesis fails. By reduction, this transfers to a bound of O(kd−1−e) 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−e) that we can prove.Most of our lower bound proofs follow a general scheme that we discover: To exclude kernels of size O(kd−e) 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.
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Citations
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Kernelization Lower Bounds By Cross-Composition
TL;DR: In this paper, the authors introduce the cross-composition framework for proving kernelization lower bounds, which generalizes and strengthens the recent techniques of using composition algorithms and of transferring the lower bounds via polynomial parameter transformations.
Parameterized Algorithms
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TL;DR: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.
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Satisfiability Allows No Nontrivial Sparsification unless the Polynomial-Time Hierarchy Collapses
Holger Dell,Dieter van Melkebeek +1 more
TL;DR: Under the hypothesis that coNP is not in NP/poly, the result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs.
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Recent developments in kernelization: A survey
TL;DR: This survey gives a general introduction to the area of kernelization and discusses some recent developments, and attempts a reasonably self-contained update and introduction on the following topics: Lower bounds for kernelization, taking into account the recent progress on the and-conjecture.
Weak compositions and their applications to polynomial lower bounds for kernelization
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- 17 Jan 2012
TL;DR: In this article, weak composition was used to obtain polynomial kernelization lower bounds for several natural parameterized problems, such as d-Bipartite Regular Perfect Code and d-Dimensional Matching.
88
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