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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| Published in: | Theoretical computer science Vol. 412; no. 35; pp. 4570 - 4578 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Oxford
Elsevier B.V
12.08.2011
Elsevier |
| Subjects: | |
| ISSN: | 0304-3975, 1879-2294 |
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| Abstract | 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
)
. |
|---|---|
| AbstractList | 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
)
. 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 ) . In this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P aS 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. and Fortnow and Santhanam , that show that NP-complete problems that are 'or-compositional' do not have polynomial kernels, unless N P aS 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 aS 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 aS 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 ) . |
| Author | Thomassé, Stéphan Bodlaender, Hans L. Yeo, Anders |
| Author_xml | – sequence: 1 givenname: Hans L. surname: Bodlaender fullname: Bodlaender, Hans L. email: hansb@cs.uu.nl organization: Department of Computer Science, Utrecht University, 3508 TB Utrecht, Netherlands – sequence: 2 givenname: Stéphan surname: Thomassé fullname: Thomassé, Stéphan email: thomasse@lirmm.fr organization: LIRMM-Université Montpellier II, 161 Rue Ada, 34392 Montpellier Cedex, France – sequence: 3 givenname: Anders surname: Yeo fullname: Yeo, Anders email: anders@cs.rhul.ac.uk organization: Department of Computer Science, Royal Holloway University of London, Egham, Surrey TW20 OEX, United Kingdom |
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| Cites_doi | 10.1007/s00224-010-9270-y 10.1006/jctb.1995.1006 10.1016/S0166-218X(03)00440-2 10.1137/S0097539792228228 10.1007/s00453-008-9180-4 10.1016/j.jcss.2010.06.007 10.1145/1721837.1721848 10.1016/j.jcss.2009.04.001 10.1007/s00224-009-9234-2 10.1145/1233481.1233493 10.1145/1806689.1806725 10.1007/s003730200002 10.1002/net.1975.5.1.45 10.1016/0304-3975(94)00097-3 |
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| Issue | 35 |
| Keywords | 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 |
| Language | English |
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| Snippet | 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... In this paper, we show that the problems Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless N P aS c o N P / p o l y . Thus, these... 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... |
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| SubjectTerms | 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 |
| Title | Kernel bounds for disjoint cycles and disjoint paths |
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