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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Bibliographic Details
Published in:Theoretical computer science Vol. 412; no. 35; pp. 4570 - 4578
Main Authors: Bodlaender, Hans L., Thomassé, Stéphan, Yeo, Anders
Format: Journal Article
Language:English
Published: Oxford Elsevier B.V 12.08.2011
Elsevier
Subjects:
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
Online Access:Get full text
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Summary: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 ) .
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2011.04.039