Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor
The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k - Path admits a polynomial Tur...
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| Veröffentlicht in: | Algorithmica Jg. 81; H. 10; S. 3936 - 3967 |
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01.10.2019
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| Abstract | The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether
k
-
Path
admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length
k
, using an oracle that answers queries of size
k
O
(
1
)
? We show this can be done when the input graph avoids a fixed graph
H
as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and
K
3
,
t
-minor-free graphs. Moreover, we show that
k
-
Path
even admits a polynomial Turing kernel when the input graph is not
H
-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any
H
-topological-minor-free graph that does not contain a
k
-path, has a separation that can safely be reduced after communication with the oracle. |
|---|---|
| AbstractList | The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether
k
-
Path
admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length
k
, using an oracle that answers queries of size
k
O
(
1
)
? We show this can be done when the input graph avoids a fixed graph
H
as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and
K
3
,
t
-minor-free graphs. Moreover, we show that
k
-
Path
even admits a polynomial Turing kernel when the input graph is not
H
-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any
H
-topological-minor-free graph that does not contain a
k
-path, has a separation that can safely be reduced after communication with the oracle. The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size kO(1)? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K3,t-minor-free graphs. Moreover, we show that k-Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle. |
| Author | Jansen, Bart M. P. Pilipczuk, Marcin Wrochna, Marcin |
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| References | ChenGYuXLong cycles in 3-connected graphsJ. Comb. Theory Ser. B20028618099193012410.1006/jctb.2002.21131025.05036 Ambalath, A.M., Balasundaram, R.,Rao, R., Koppula, V., Misra, N., Philip, G., Ramanujan, M.S.: On the kernelization complexity of colorful motifs. In: Proceedings of the 5th IPEC, pp. 14–25 (2010). https://doi.org/10.1007/978-3-642-17493-3_4 GajarskýJHlinenýPObdrzálekJOrdyniakSReidlFRossmanithPVillaamilFSSikdarSKernelization using structural parameters on sparse graph classesJ. Comput. Syst. Sci.201784219242357017810.1016/j.jcss.2016.09.0021353.68127 Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-27875-4 Shan, S.: Homeomorphically irreducible spanning trees, Halin graphs, and long cycles in 3-connected graphs with bounded maximum degrees. Ph.D. thesis, Georgia State University (2015). http://scholarworks.gsu.edu/math_diss/23/. Accessed 5 Nov 2015 ThomasséSTrotignonNVuskovicKA polynomial turing-kernel for weighted independent set in bull-free graphsAlgorithmica2017773619641360438710.1007/s00453-015-0083-x1364.68233 ChenGYuXZangWThe circumference of a graph with no K3,t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{3, t}$$\end{document}-minor, IIJ. Comb. Theory Ser. B201210261211124010.1016/j.jctb.2012.07.0031256.05228 ArchdeaconDTopological graph theory: a surveyCongr. Numerantium19961155–541814112360897.05026 Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Minimum bisection is fixed parameter tractable. In: Proceedings of STOC 2014, pp. 323–332. ACM (2014). https://doi.org/10.1145/2591796.2591852 GroheMMarxDStructure theorem and isomorphism test for graphs with excluded topological subgraphsSIAM J. Comput.2015441114159331356910.1137/1208922341314.05134 Hüffner, F., Komusiewicz, C., Sorge, M.: Finding highly connected subgraphs. In: Proceedings of 41st SOFSEM, pp. 254–265 (2015). https://doi.org/10.1007/978-3-662-46078-8_21 Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Manuscript (2019) Kolay, S., Panolan, F.: Parameterized algorithms for deletion to (r,ℓ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r, \ell )$$\end{document}-graphs. In: Proceedings of 35th FSTTCS, pp. 420–433 (2015). https://doi.org/10.4230/LIPIcs.FSTTCS.2015.420 BarberoFPaulCPilipczukMExploring the complexity of layout parameters in tournaments and semicomplete digraphsACM Trans. Algorithms201814338:138:31384134010.1145/319627606979228 SchäferAKomusiewiczCMoserHNiedermeierRParameterized computational complexity of finding small-diameter subgraphsOptim. Lett.201265883891292562410.1007/s11590-011-0311-51254.90279 Jansen, B.M.P., Marx, D.: Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels. In: Proceedings of 26th SODA, pp. 616–629 (2015). https://doi.org/10.1137/1.9781611973730.42 DiestelRKawarabayashiKMüllerTWollanPOn the excluded minor structure theorem for graphs of large tree-widthJ. Comb. Theory Ser. B2012102611891210299297610.1016/j.jctb.2012.07.0011256.05229 JansenBMPTuring kernelization for finding long paths and cycles in restricted graph classesJ. Comput. Syst. Sci.2017851837358410910.1016/j.jcss.2016.10.0081356.68099 FröhlichJMüllerTLinear connectivity forces large complete bipartite minors: an alternative approachJ. Comb. Theory Ser. B20111016502508283281610.1016/j.jctb.2011.02.0021234.05221 RobertsonNSeymourPDGraph minors. V. Excluding a planar graphJ. Comb. Theory Ser. B19864119211485460610.1016/0095-8956(86)90030-40598.05055 GiacomoEDLiottaGMchedlidzeTLower and upper bounds for long induced paths in 3-connected planar graphsTheor. Comput. Sci.20166364755350653810.1016/j.tcs.2016.04.0341342.05071 Binkele-RaibleDFernauHFominFVLokshtanovDSaurabhSVillangerYKernel(s) for problems with no kernel: on out-trees with many leavesACM Trans. Algorithms20128438298191610.1145/2344422.23444281295.68120 DiestelRGraph Theory20104HeidelbergSpringer10.1007/978-3-642-14279-61204.05001 RobertsonNSeymourPDGraph minors: XVII. Taming a vortexJ. Comb. Theory Ser. B1999771162210171053810.1006/jctb.1999.19191027.05088 RobertsonNSeymourPDGraph minors. X. Obstructions to tree-decompositionJ. Comb. Theory Ser. B1991522153190111046810.1016/0095-8956(91)90061-N0764.05069 Bodlaender, H.L., Demaine, E.D., Fellows, M.R., Guo, J., Hermelin, D., Lokshtanov, D., Müller, M., Raman, V., Rooij, J.V., Rosamond, F.A.: Open problems in parameterized and exact computation—IWPEC 2008. Technical Report UU-CS-2008-017, Utrecht University (2008) BodlaenderHLJansenBMPKratschSKernelization lower bounds by cross-compositionSIAM J. Discrete Math.2014281277305316695610.1137/1208802401295.05222 DiracGASome theorems on abstract graphsProc. Lond. Math. Soc.1952s3–2169814730810.1112/plms/s3-2.1.690047.17001 TutteWTConnectivity in Graphs. Mathematical expositions1966TorontoUniversity of Toronto Press ChenGGaoZYuXZangWApproximating longest cycles in graphs with bounded degreesSIAM J. Comput.2006363635656226300510.1137/0506332631118.05047 HermelinDKratschSSoltysKWahlströmMWuXA completeness theory for polynomial (Turing) kernelizationAlgorithmica2015713702730331524410.1007/s00453-014-9910-81312.68102 GarneroVWellerMParameterized certificate dispersal and its variantsTheor. Comput. Sci.20166226678346183210.1016/j.tcs.2016.02.0011335.68104 WT Tutte (614_CR32) 1966 S Thomassé (614_CR31) 2017; 77 N Robertson (614_CR27) 1991; 52 HL Bodlaender (614_CR6) 2014; 28 614_CR1 BMP Jansen (614_CR21) 2017; 85 J Fröhlich (614_CR14) 2011; 101 ED Giacomo (614_CR17) 2016; 636 D Hermelin (614_CR19) 2015; 71 V Garnero (614_CR16) 2016; 622 D Binkele-Raible (614_CR4) 2012; 8 N Robertson (614_CR28) 1999; 77 N Robertson (614_CR26) 1986; 41 614_CR10 R Diestel (614_CR11) 2010 614_CR30 M Grohe (614_CR18) 2015; 44 J Gajarský (614_CR15) 2017; 84 D Archdeacon (614_CR2) 1996; 115 A Schäfer (614_CR29) 2012; 6 GA Dirac (614_CR13) 1952; s3–2 R Diestel (614_CR12) 2012; 102 614_CR24 614_CR23 614_CR22 614_CR20 614_CR5 F Barbero (614_CR3) 2018; 14 G Chen (614_CR9) 2012; 102 G Chen (614_CR8) 2002; 86 G Chen (614_CR7) 2006; 36 614_CR25 |
| References_xml | – reference: BarberoFPaulCPilipczukMExploring the complexity of layout parameters in tournaments and semicomplete digraphsACM Trans. Algorithms201814338:138:31384134010.1145/319627606979228 – reference: GarneroVWellerMParameterized certificate dispersal and its variantsTheor. Comput. Sci.20166226678346183210.1016/j.tcs.2016.02.0011335.68104 – reference: ArchdeaconDTopological graph theory: a surveyCongr. Numerantium19961155–541814112360897.05026 – reference: Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-27875-4 – reference: DiracGASome theorems on abstract graphsProc. Lond. Math. Soc.1952s3–2169814730810.1112/plms/s3-2.1.690047.17001 – reference: Jansen, B.M.P., Marx, D.: Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels. In: Proceedings of 26th SODA, pp. 616–629 (2015). https://doi.org/10.1137/1.9781611973730.42 – reference: ThomasséSTrotignonNVuskovicKA polynomial turing-kernel for weighted independent set in bull-free graphsAlgorithmica2017773619641360438710.1007/s00453-015-0083-x1364.68233 – reference: DiestelRKawarabayashiKMüllerTWollanPOn the excluded minor structure theorem for graphs of large tree-widthJ. Comb. Theory Ser. B2012102611891210299297610.1016/j.jctb.2012.07.0011256.05229 – reference: SchäferAKomusiewiczCMoserHNiedermeierRParameterized computational complexity of finding small-diameter subgraphsOptim. Lett.201265883891292562410.1007/s11590-011-0311-51254.90279 – reference: BodlaenderHLJansenBMPKratschSKernelization lower bounds by cross-compositionSIAM J. Discrete Math.2014281277305316695610.1137/1208802401295.05222 – reference: RobertsonNSeymourPDGraph minors: XVII. Taming a vortexJ. Comb. Theory Ser. B1999771162210171053810.1006/jctb.1999.19191027.05088 – reference: GajarskýJHlinenýPObdrzálekJOrdyniakSReidlFRossmanithPVillaamilFSSikdarSKernelization using structural parameters on sparse graph classesJ. Comput. Syst. Sci.201784219242357017810.1016/j.jcss.2016.09.0021353.68127 – reference: Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Manuscript (2019) – reference: HermelinDKratschSSoltysKWahlströmMWuXA completeness theory for polynomial (Turing) kernelizationAlgorithmica2015713702730331524410.1007/s00453-014-9910-81312.68102 – reference: DiestelRGraph Theory20104HeidelbergSpringer10.1007/978-3-642-14279-61204.05001 – reference: GroheMMarxDStructure theorem and isomorphism test for graphs with excluded topological subgraphsSIAM J. Comput.2015441114159331356910.1137/1208922341314.05134 – reference: ChenGYuXLong cycles in 3-connected graphsJ. Comb. Theory Ser. 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| Title | Turing Kernelization for Finding Long Paths in Graph Classes Excluding a Topological Minor |
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