Kernel bounds for disjoint cycles and disjoint paths

In this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . Thus, these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand. We bu...

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Veröffentlicht in:	Theoretical computer science Jg. 412; H. 35; S. 4570 - 4578
Hauptverfasser:	Bodlaender, Hans L., Thomassé, Stéphan, Yeo, Anders
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Oxford Elsevier B.V 12.08.2011 Elsevier
Schlagworte:	Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences C (programming language) Combinatorics Combinatorics. Ordered structures Computer Science Computer science; control theory; systems Construction Data reduction Designs and configurations Discrete Mathematics Disjoint cycles Exact sciences and technology Fixed parameter complexity Kernelization Kernels Lower bounds Machinery Mathematics Miscellaneous Preprocessing Proving Sciences and techniques of general use Strings Theoretical computing Transformations Lower bounds Fixed parameter complexity Disjoint cycles Algorithms Data reduction Kernelization Polynomial Lower bound Computer theory Time allowed Algorithm Complexity Polynomial time Packing Proof NP complete problem
ISSN:	0304-3975, 1879-2294
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Zusammenfassung:	In this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . Thus, these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand. We build upon recent results by Bodlaender et al. [6] and Fortnow and Santhanam [20], that show that NP-complete problems that are ‘or-compositional’ do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . To this machinery, we add a notion of transformation, and obtain that Disjoint Cycles, and Disjoint Paths do not have polynomial kernels, unless N P ⊆ c o N P / p o l y . For the proof, we introduce a problem on strings, called Disjoint Factors, and first show that this problem has no polynomial kernel unless N P ⊆ c o N P / p o l y . We also show that the related Disjoint Cycles Packing problem has a kernel of size O ( k log k ) .
Bibliographie:	ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0304-3975 1879-2294
DOI:	10.1016/j.tcs.2011.04.039