Kernels for FEEDBACK ARC SET IN TOURNAMENTS

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Publié par

Kernels for FEEDBACK ARC SET IN TOURNAMENTS Anthony Perez Joint work with S. Bessy, F. V. Fomin, S. Gaspers, C. Paul, S. Saurabh, S. Thomassé Université Montpellier II - LIRMM JGA'09 - Montpellier A. Perez (LIRMM) KERNELS FOR k -FAST 06 Novembre 2009 1 / 16

  • depends only

  • feedback arc

  • parameterized algorithms

  • parameterized algorithm

  • exact resolution - parameterized

  • results reduction


Publié le : mardi 19 juin 2012
Lecture(s) : 19
Source : lirmm.fr
Nombre de pages : 24
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zereRIL(P.AFOLS-FRk)KMMNEER2e00merbN6voSA0T
Université Montpellier II - LIRMM
JGA’09 - Montpellier
6
Kernels for FCAKEEDBARCSET INTENAMTSUONR
Anthony Perez Joint work with S. Bessy, F. V. Fomin, S. Gaspers, C. Paul, S. Saurabh, S. Thomassé
91/1
RkFOAS-FER)KLSNEL(zeMMRIAreP.
Kernels fork-FAST Definitions and structural results Reduction rules and size
3
Conclusion
/16
Plan
Exact resolution - parameterized algorithms
1
2
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neP.A(zer
Conclusion
3
Kernels fork-FAST Definitions and structural results Reduction rules and size
2
Exact resolution - parameterized algorithms
1
61/39002FSRONRLE)MEKILMRmbreNoveST06k-FAsPlaithmlgorzedaetiraremaP
mhtisadezroglrameteriPa94/16
Parameterized algorithm A problem parameterized bykNis said to be fixed-parameter tractable (inFPT) if it can be solved in timef(k).nO(1).
Remarks The functionfconsidered can be anything and depends only on the parameterk. Thus, the functionf(k) =222kis good.
embre200SA0TN6voSLOFkRF-)KMMNEERezerIR(LP.A
deziretemaraPlaogirhtsmLEFSEKNRMR)M(zILPereA.20re4/09
Parameterized algorithm A problem parameterized bykNis said to be fixed-parameter ) tractable (inFPT) if it can be solved in timef(k).nO(1.
16
Remarks The functionfconsidered can be anything and depends only on the parameterk. Thus, the functionf(k) =222kis good.
-kROTSAFoN60bmev
msthrigodelarezimatePra
Kernelization Given a parameterized problemΠand(x,k)Π, akernelizationis a polynomial-time algorithm(set of reduction rules)that takes as input (x,k)Πand outputs(x0,k0)Πs.t. :
61
Theorem ΠFPTΠhas a kernel (size : exponential).
xis a YES-instancex0is a YES-instance |x0| ≤h(k) k0k
20re5/09No06mbveAFTSRO-kLEFSEKNRRMM)z(LIPereA.
aParemetirezlgdaitorshmA.Perez(LIRMMK)REENSLOFkRF-ov6NT0AS00e2brem
xis a YES-instancex0is a YES-instance |x0| ≤h(k) k0k
Kernelization Given a parameterized problemΠand(x,k)Π, akernelizationis a polynomial-time algorithm(set of reduction rules)that takes as input (x,k)Πand outputs(x0,k0)Πs.t. :
61/59
Theorem ΠFPTΠhas a kernel (size : exponential).
No06STFAk-ORSFELNREK)MMRIL(zerePA.90/661
Consequences Pre-processing Reducing the size of a given input Resolution on kernels Additive complexity (O(g(k) +poly(n))).
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rez(A.Pe
1
2
Exact resolution - parameterized algorithms
3
Kernels fork-FAST Definitions and structural results Reduction rules and size
Conclusion
0290/761LENRROFSMRILEK)MveNorembFAk-06STtionenistrusandlaertcruPsalustlKofslenreDTSAF-krn
ultsnrlefsroeKarutserlsdnacurtitnnsioFAk-DeST
Tournament Atournamentis a graph obtained by orienting every edge of a complete graph. Atournamentis acyclic iff it has a transitive ordering of its vertices. Afeedback arc setfor a tournamentT= (V,A)is a (minimum) set of arcs whose removal makesTacyclic.
168/0920rembve.ARMM)KERNPerez(LIAFTS60oNLEFSRO-k
0290bmer
Lemma (Raman, Saurabh, TCS 2006) LetD= (V,A)be a digraph andFa feedback arc set forD. The graph D0obtained by reversing all arcs ofFinDis acyclic.
FKEEDBACARCSET INT
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