Reconfiguration of Spanning Trees with Degree Constraints or Diameter Constraints

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constrai...

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Published in:	Algorithmica Vol. 85; no. 9; pp. 2779 - 2816
Main Authors:	Bousquet, Nicolas, Ito, Takehiro, Kobayashi, Yusuke, Mizuta, Haruka, Ouvrard, Paul, Suzuki, Akira, Wasa, Kunihiro
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2023 Springer Nature B.V Springer Verlag
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Lower bounds Mathematics of Computing Polynomials Reconfiguration Theory of Computation Transformations Upper bounds Combinatorial reconfiguration Polynomial-time algorithms PSPACE Spanning trees
ISSN:	0178-4617, 1432-0541
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Abstract	We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.
AbstractList	We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.
Author	Mizuta, Haruka Ouvrard, Paul Ito, Takehiro Wasa, Kunihiro Bousquet, Nicolas Kobayashi, Yusuke Suzuki, Akira
Author_xml	– sequence: 1 givenname: Nicolas surname: Bousquet fullname: Bousquet, Nicolas organization: CNRS, LIRIS, Université de Lyon – sequence: 2 givenname: Takehiro surname: Ito fullname: Ito, Takehiro organization: Graduate School of Information Sciences, Tohoku University – sequence: 3 givenname: Yusuke surname: Kobayashi fullname: Kobayashi, Yusuke organization: Research Institute for Mathematical Sciences, Kyoto University – sequence: 4 givenname: Haruka surname: Mizuta fullname: Mizuta, Haruka organization: Graduate School of Information Sciences, Tohoku University – sequence: 5 givenname: Paul surname: Ouvrard fullname: Ouvrard, Paul organization: Université de Bordeaux – sequence: 6 givenname: Akira orcidid: 0000-0002-5212-0202 surname: Suzuki fullname: Suzuki, Akira email: akira@tohoku.ac.jp organization: Graduate School of Information Sciences, Tohoku University – sequence: 7 givenname: Kunihiro surname: Wasa fullname: Wasa, Kunihiro organization: Faculty of Science and Engineering, Hosei University
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References	SpriggsMJKeilJMBespamyatnikhSSegalMSnoeyinkJComputing a (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\varepsilon )$$\end{document}-approximate geometric minimum-diameter spanning treeAlgorithmica2004384577589203252610.1007/s00453-003-1056-z1138.68477 Bousquet, N., Ito, T., Kobayashi, Y., Mizuta, H., Ouvrard, P., Suzuki, A., Wasa, K.: Reconfiguration of spanning trees with many or few leaves. In: 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), pp. 24:1–24:15 (2020). https://doi.org/10.4230/LIPIcs.ESA.2020.24 HassinRefaelTamirArieOn the minimum diameter spanning tree problemInf. Process. Lett.1995532109111131471710.1016/0020-0190(94)00183-Y0875.68442 Huang, S., Fu, AWC., Liu, R.: Minimum spanning trees in temporal graphs. In Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data, SIGMOD ’15, pp 419-430, New York, NY, USA, (2015). Association for Computing Machinery. https://doi.org/10.1145/2723372.2723717 ItalianoGiuseppe FRamaswamiRajivMaintaining spanning trees of small diameterAlgorithmica1998223275304164611010.1007/PL000092250915.68084 Korte, B. H., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer-Verlag, New York, NY, (2012). https://doi.org/10.1007/978-3-642-24488-9 ItoTDemaineEHarveyNJAPapadimitriouCHMarthaSideriRUnoYOn the complexity of reconfiguration problemsTheor. Comput. Sci.20114121210541065279774810.1016/j.tcs.2010.12.0051207.68166 Mizuta, H., Hatanaka, T., Ito, T., Zhou, X.: Reconfiguration of minimum steiner trees via vertex exchanges. In: 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, pp. 79:1–79:11, (2019). https://doi.org/10.4230/LIPIcs.MFCS.2019.79 Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer (2003) van den Heuvel, J.: The complexity of change. In: Simon, R., Blackburn, S.G., Mark W. (eds.)Surveys in Combinatorics, Vol. 409 of London Mathematical Society Lecture Note Series, pp. 127–160. Cambridge University Press (2013). https://doi.org/10.1017/CBO9781139506748.005 NishimuraNaomiIntroduction to reconfigurationAlgorithms201811452379890310.3390/a110400521461.68164 Singh, M., Lau, L. C.: Approximating minimum bounded degree spanning trees to within one of optimal. J. ACM, 62(1), (March 2015). https://doi.org/10.1145/2629366 Goemans, MX.: Minimum bounded degree spanning trees. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pp. 273–282. IEEE Computer Society. https://doi.org/10.1109/FOCS.2006.48 (2006) HassinRLevinAMinimum restricted diameter spanning treesDiscret. Appl. Math.20041373343357203663710.1016/S0166-218X(03)00360-31095.90094 FurerMartinRaghavachariBalajiApproximating the minimum-degree steiner tree to within one of optimalJ. Algorithms1994173409423130025710.1006/jagm.1994.10421321.05262 CabelloSergioChambersErin WEricksonJeffMultiple-source shortest paths in embedded graphsSIAM J. Comput.201342415421571307932510.1137/1208642711276.05031 Karp, R. M.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972) Czumaj, A., Strothmann, WB.: Bounded degree spanning trees. In: Rainer, B., Gerhard, W., (eds.) Algorithms — ESA ’97, Vol. 1284 of LNCS, pp. 104–117. Springer Berlin Heidelberg, Berlin Heidelberg (1997) HearnRobert ADemaineErik DPSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computationTheor. Comput. Sci.20053431–27296216884510.1016/j.tcs.2005.05.0081079.68040 Barjon, M., Casteigts, A., Chaumette, S., Johnen, C., Neggaz, Y.M.: Maintaining a spanning forest in highly dynamic networks: the synchronous case. In: Marcos K., Aguilera, L.Q., Marc S., (eds.) Principles of Distributed Systems, pp. 277–292. Springer International Publishing, Cham (2014) TakaokaAComplexity of Hamiltonian cycle reconfigurationAlgorithms2018119140386321910.3390/a110901401461.68166 Martin Furer (1117_CR5) 1994; 17 Robert A Hearn (1117_CR9) 2005; 343 T Ito (1117_CR13) 2011; 412 R Hassin (1117_CR7) 2004; 137 Refael Hassin (1117_CR8) 1995; 53 Sergio Cabello (1117_CR3) 2013; 42 1117_CR19 MJ Spriggs (1117_CR20) 2004; 38 A Takaoka (1117_CR21) 2018; 11 1117_CR2 1117_CR11 1117_CR1 1117_CR14 Naomi Nishimura (1117_CR17) 2018; 11 1117_CR16 1117_CR6 1117_CR15 1117_CR18 1117_CR4 Giuseppe F Italiano (1117_CR12) 1998; 22 1117_CR10
References_xml	– reference: Mizuta, H., Hatanaka, T., Ito, T., Zhou, X.: Reconfiguration of minimum steiner trees via vertex exchanges. In: 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, pp. 79:1–79:11, (2019). https://doi.org/10.4230/LIPIcs.MFCS.2019.79 – reference: HearnRobert ADemaineErik DPSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computationTheor. Comput. Sci.20053431–27296216884510.1016/j.tcs.2005.05.0081079.68040 – reference: CabelloSergioChambersErin WEricksonJeffMultiple-source shortest paths in embedded graphsSIAM J. Comput.201342415421571307932510.1137/1208642711276.05031 – reference: Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer (2003) – reference: SpriggsMJKeilJMBespamyatnikhSSegalMSnoeyinkJComputing a (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\varepsilon )$$\end{document}-approximate geometric minimum-diameter spanning treeAlgorithmica2004384577589203252610.1007/s00453-003-1056-z1138.68477 – reference: FurerMartinRaghavachariBalajiApproximating the minimum-degree steiner tree to within one of optimalJ. Algorithms1994173409423130025710.1006/jagm.1994.10421321.05262 – reference: TakaokaAComplexity of Hamiltonian cycle reconfigurationAlgorithms2018119140386321910.3390/a110901401461.68166 – reference: ItoTDemaineEHarveyNJAPapadimitriouCHMarthaSideriRUnoYOn the complexity of reconfiguration problemsTheor. Comput. Sci.20114121210541065279774810.1016/j.tcs.2010.12.0051207.68166 – reference: Korte, B. H., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer-Verlag, New York, NY, (2012). https://doi.org/10.1007/978-3-642-24488-9 – reference: Goemans, MX.: Minimum bounded degree spanning trees. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pp. 273–282. IEEE Computer Society. https://doi.org/10.1109/FOCS.2006.48 (2006) – reference: HassinRefaelTamirArieOn the minimum diameter spanning tree problemInf. Process. Lett.1995532109111131471710.1016/0020-0190(94)00183-Y0875.68442 – reference: Karp, R. M.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972) – reference: van den Heuvel, J.: The complexity of change. In: Simon, R., Blackburn, S.G., Mark W. (eds.)Surveys in Combinatorics, Vol. 409 of London Mathematical Society Lecture Note Series, pp. 127–160. Cambridge University Press (2013). https://doi.org/10.1017/CBO9781139506748.005 – reference: Barjon, M., Casteigts, A., Chaumette, S., Johnen, C., Neggaz, Y.M.: Maintaining a spanning forest in highly dynamic networks: the synchronous case. In: Marcos K., Aguilera, L.Q., Marc S., (eds.) Principles of Distributed Systems, pp. 277–292. Springer International Publishing, Cham (2014) – reference: Bousquet, N., Ito, T., Kobayashi, Y., Mizuta, H., Ouvrard, P., Suzuki, A., Wasa, K.: Reconfiguration of spanning trees with many or few leaves. In: 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), pp. 24:1–24:15 (2020). https://doi.org/10.4230/LIPIcs.ESA.2020.24 – reference: Czumaj, A., Strothmann, WB.: Bounded degree spanning trees. In: Rainer, B., Gerhard, W., (eds.) Algorithms — ESA ’97, Vol. 1284 of LNCS, pp. 104–117. Springer Berlin Heidelberg, Berlin Heidelberg (1997) – reference: Singh, M., Lau, L. C.: Approximating minimum bounded degree spanning trees to within one of optimal. J. ACM, 62(1), (March 2015). https://doi.org/10.1145/2629366 – reference: NishimuraNaomiIntroduction to reconfigurationAlgorithms201811452379890310.3390/a110400521461.68164 – reference: Huang, S., Fu, AWC., Liu, R.: Minimum spanning trees in temporal graphs. In Proceedings of the 2015 ACM SIGMOD International Conference on Management of Data, SIGMOD ’15, pp 419-430, New York, NY, USA, (2015). Association for Computing Machinery. https://doi.org/10.1145/2723372.2723717 – reference: HassinRLevinAMinimum restricted diameter spanning treesDiscret. Appl. Math.20041373343357203663710.1016/S0166-218X(03)00360-31095.90094 – reference: ItalianoGiuseppe FRamaswamiRajivMaintaining spanning trees of small diameterAlgorithmica1998223275304164611010.1007/PL000092250915.68084 – ident: 1117_CR15 doi: 10.1007/978-3-642-24488-9 – volume: 137 start-page: 343 issue: 3 year: 2004 ident: 1117_CR7 publication-title: Discret. 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Algorithms doi: 10.1006/jagm.1994.1042 – ident: 1117_CR6 doi: 10.1109/FOCS.2006.48 – ident: 1117_CR11 doi: 10.1145/2723372.2723717 – ident: 1117_CR1 doi: 10.1007/978-3-319-14472-6_19 – ident: 1117_CR4 doi: 10.1007/3-540-63397-9_9 – volume: 53 start-page: 109 issue: 2 year: 1995 ident: 1117_CR8 publication-title: Inf. Process. Lett. doi: 10.1016/0020-0190(94)00183-Y – volume: 343 start-page: 72 issue: 1–2 year: 2005 ident: 1117_CR9 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2005.05.008 – volume: 42 start-page: 1542 issue: 4 year: 2013 ident: 1117_CR3 publication-title: SIAM J. Comput. doi: 10.1137/120864271 – ident: 1117_CR14 doi: 10.1007/978-1-4684-2001-2_9 – ident: 1117_CR18
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Lower bounds Mathematics of Computing Polynomials Reconfiguration Theory of Computation Transformations Upper bounds
Title	Reconfiguration of Spanning Trees with Degree Constraints or Diameter Constraints
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