Bounded fixed-parameter tractability and reducibility
We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family ℱ . For every family ℱ of functions, this yields a notion of ℱ - fixed-parameter tractability. If ℱ is...
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| Veröffentlicht in: | Annals of pure and applied logic Jg. 148; H. 1; S. 1 - 19 |
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| Hauptverfasser: | , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Elsevier B.V
01.09.2007
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| Schlagworte: | |
| ISSN: | 0168-0072 |
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| Abstract | We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family
ℱ
. For every family
ℱ
of functions, this yields a notion of
ℱ
-
fixed-parameter tractability. If
ℱ
is the class of all polynomially bounded functions, then
ℱ
-fixed-parameter tractability coincides with polynomial time decidability and if
ℱ
is the class of all computable functions,
ℱ
-fixed-parameter tractability coincides with the standard notion of fixed-parameter tractability. There are interesting choices of
ℱ
between these two extremes, for example the class of all singly exponential functions.
In this article, we study the general theory of
ℱ
-fixed-parameter tractability. We introduce a generic notion of reduction and two classes
ℱ
-W[P]
and
ℱ
-XP
, which may be viewed as analogues of NP and EXPTIME, respectively, in the world of
ℱ
-fixed-parameter tractability. |
|---|---|
| AbstractList | We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family
ℱ
. For every family
ℱ
of functions, this yields a notion of
ℱ
-
fixed-parameter tractability. If
ℱ
is the class of all polynomially bounded functions, then
ℱ
-fixed-parameter tractability coincides with polynomial time decidability and if
ℱ
is the class of all computable functions,
ℱ
-fixed-parameter tractability coincides with the standard notion of fixed-parameter tractability. There are interesting choices of
ℱ
between these two extremes, for example the class of all singly exponential functions.
In this article, we study the general theory of
ℱ
-fixed-parameter tractability. We introduce a generic notion of reduction and two classes
ℱ
-W[P]
and
ℱ
-XP
, which may be viewed as analogues of NP and EXPTIME, respectively, in the world of
ℱ
-fixed-parameter tractability. |
| Author | Downey, Rod Flum, Jörg Weyer, Mark Grohe, Martin |
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| Keywords | Reductions Parameterized complexity theory |
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| References | Abrahamson, Downey, Fellows (b1) 1995; 73 Chen, Flum, Grohe (b5) 2005; 339 Cutland (b7) 1980 Frick, Grohe (b11) 2004; 130 Flum, Grohe, Weyer (b10) 2006; 72 Weyer (b14) 2004; vol. 3162 Kintala, Fischer (b12) 1980; 9 Cai, Chen (b3) 1997; 26 Cai, Chen (b2) 1997; 54 Y. Chen, M. Grohe, An isomorphism between subexponential and parameterized complexity theory, in: Proceedings of the 21st Conference on Computational Complexity, 2006, pp. 314–328 Flum, Grohe (b9) 2006 Downey, Fellows (b8) 1999 Cai, Chen, Downey, Fellows (b4) 1995; 123 Robertson, Seymour (b13) 1995; 63 Frick (10.1016/j.apal.2007.06.001_b11) 2004; 130 Cai (10.1016/j.apal.2007.06.001_b3) 1997; 26 Cai (10.1016/j.apal.2007.06.001_b4) 1995; 123 Chen (10.1016/j.apal.2007.06.001_b5) 2005; 339 Kintala (10.1016/j.apal.2007.06.001_b12) 1980; 9 Abrahamson (10.1016/j.apal.2007.06.001_b1) 1995; 73 Cutland (10.1016/j.apal.2007.06.001_b7) 1980 Robertson (10.1016/j.apal.2007.06.001_b13) 1995; 63 Weyer (10.1016/j.apal.2007.06.001_b14) 2004; vol. 3162 Cai (10.1016/j.apal.2007.06.001_b2) 1997; 54 10.1016/j.apal.2007.06.001_b6 Downey (10.1016/j.apal.2007.06.001_b8) 1999 Flum (10.1016/j.apal.2007.06.001_b9) 2006 Flum (10.1016/j.apal.2007.06.001_b10) 2006; 72 |
| References_xml | – year: 1980 ident: b7 article-title: Computability – volume: 130 start-page: 3 year: 2004 end-page: 31 ident: b11 article-title: The complexity of first-order and monadic second-order logic revisited publication-title: Annals of Pure and Applied Logic – volume: 73 start-page: 235 year: 1995 end-page: 276 ident: b1 article-title: Fixed-parameter tractability and completeness IV: On completeness for W[P] and PSPACE analogs publication-title: Annals of Pure and Applied Logic – volume: 9 start-page: 46 year: 1980 end-page: 53 ident: b12 article-title: Refining nondeterminism in relativised polynomial time bounded computations publication-title: SIAM Journal on Computing – reference: Y. Chen, M. Grohe, An isomorphism between subexponential and parameterized complexity theory, in: Proceedings of the 21st Conference on Computational Complexity, 2006, pp. 314–328 – year: 1999 ident: b8 article-title: Parameterized Complexity – volume: 72 start-page: 34 year: 2006 end-page: 71 ident: b10 article-title: Bounded fixed-parameter tractability and publication-title: Journal of Computer and System Sciences – volume: 26 start-page: 733 year: 1997 end-page: 750 ident: b3 article-title: On the amount of nondeterminism and the power of verifying publication-title: SIAM Journal on Computing – year: 2006 ident: b9 article-title: Parameterized Complexity Theory – volume: 339 start-page: 167 year: 2005 end-page: 199 ident: b5 article-title: Machine-based methods in parameterized complexity theory publication-title: Theoretical Computer Science – volume: 54 start-page: 465 year: 1997 end-page: 474 ident: b2 article-title: On fixed-parameter tractability and approximability of NP optimization problems publication-title: Journal of Computer and System Sciences – volume: vol. 3162 start-page: 49 year: 2004 end-page: 60 ident: b14 article-title: Bounded fixed-parameter tractability: The case publication-title: Proceedings of the 1st International Workshop on Parameterized and Exact Computation – volume: 123 start-page: 38 year: 1995 end-page: 49 ident: b4 article-title: On the structure of parameterized problems in NP publication-title: Information and Computation – volume: 63 start-page: 65 year: 1995 end-page: 110 ident: b13 article-title: Graph minors XIII. The disjoint paths problem publication-title: Journal of Combinatorial Theory, Series B – volume: 73 start-page: 235 year: 1995 ident: 10.1016/j.apal.2007.06.001_b1 article-title: Fixed-parameter tractability and completeness IV: On completeness for W[P] and PSPACE analogs publication-title: Annals of Pure and Applied Logic doi: 10.1016/0168-0072(94)00034-Z – year: 1999 ident: 10.1016/j.apal.2007.06.001_b8 – volume: 123 start-page: 38 year: 1995 ident: 10.1016/j.apal.2007.06.001_b4 article-title: On the structure of parameterized problems in NP publication-title: Information and Computation doi: 10.1006/inco.1995.1156 – year: 1980 ident: 10.1016/j.apal.2007.06.001_b7 – volume: 26 start-page: 733 issue: 3 year: 1997 ident: 10.1016/j.apal.2007.06.001_b3 article-title: On the amount of nondeterminism and the power of verifying publication-title: SIAM Journal on Computing doi: 10.1137/S0097539793258295 – volume: vol. 3162 start-page: 49 year: 2004 ident: 10.1016/j.apal.2007.06.001_b14 article-title: Bounded fixed-parameter tractability: The case 2poly(k) – volume: 63 start-page: 65 year: 1995 ident: 10.1016/j.apal.2007.06.001_b13 article-title: Graph minors XIII. The disjoint paths problem publication-title: Journal of Combinatorial Theory, Series B doi: 10.1006/jctb.1995.1006 – volume: 9 start-page: 46 year: 1980 ident: 10.1016/j.apal.2007.06.001_b12 article-title: Refining nondeterminism in relativised polynomial time bounded computations publication-title: SIAM Journal on Computing doi: 10.1137/0209003 – volume: 72 start-page: 34 year: 2006 ident: 10.1016/j.apal.2007.06.001_b10 article-title: Bounded fixed-parameter tractability and log2n nondeterministic bits publication-title: Journal of Computer and System Sciences doi: 10.1016/j.jcss.2005.06.003 – ident: 10.1016/j.apal.2007.06.001_b6 – volume: 339 start-page: 167 year: 2005 ident: 10.1016/j.apal.2007.06.001_b5 article-title: Machine-based methods in parameterized complexity theory publication-title: Theoretical Computer Science doi: 10.1016/j.tcs.2005.02.003 – volume: 54 start-page: 465 year: 1997 ident: 10.1016/j.apal.2007.06.001_b2 article-title: On fixed-parameter tractability and approximability of NP optimization problems publication-title: Journal of Computer and System Sciences doi: 10.1006/jcss.1997.1490 – year: 2006 ident: 10.1016/j.apal.2007.06.001_b9 – volume: 130 start-page: 3 year: 2004 ident: 10.1016/j.apal.2007.06.001_b11 article-title: The complexity of first-order and monadic second-order logic revisited publication-title: Annals of Pure and Applied Logic doi: 10.1016/j.apal.2004.01.007 |
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| Title | Bounded fixed-parameter tractability and reducibility |
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