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On the Approximability of the Minimum Fundamental Cycle Basis Problem

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Approximation and Online Algorithms (WAOA 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2909))

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Abstract

We consider the problem of finding a fundamental cycle basis of minimum total weight in the cycle space associated with an undirected biconnected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Although several heuristics have been proposed to tackle this NP-hard problem, which has several interesting applications, nothing is known regarding its approximability. In this paper we show that this problem is MAXSNP-hard and hence does not admit a polynomial-time approximation scheme (PTAS) unless P=NP. We also derive the first upper bounds on the approximability of the problem for arbitrary and dense graphs. In particular, for complete graphs, it is approximable within 4+ε , for any ε >0.

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References

  1. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and hardness of approximation problems. Journal of ACM 45, 501–555 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alon, N., Karp, R., Peleg, D., West, D.: Graph-theoretic game and its application to the k-server problem. SIAM J. Comput. 24, 78–100 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation: Combinatorial optimization problems and their approximability properties. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  4. Bartal, Y.: On the approximation of metric spaces by tree metrics. In: Proceedings of STOC 1998, pp. 161–168 (1998)

    Google Scholar 

  5. Brambilla, A., Premoli, A.: Rigorous event-driven (RED) analysis of large scale RC circuits. IEEE Trans. on CAS-I 48, 938–954 (2001)

    Article  Google Scholar 

  6. Charikar, M., Chekuri, C., Goel, A., Guha, S., Plotkin, S.: Approximating a finite metric by a small number of tree metrics. In: Proceedings of FOCS 1998, pp. 161–168 (1998)

    Google Scholar 

  7. Deo, N., Prabhu, G.M., Krishamoorthy, M.S.: Algorithms for generating fundamental cycles in a graph. ACM Trans. Math. Software 8, 26–42 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  8. Deo, N., Kumar, N., Parsons, J.: Minimum-length fundamental-cycle set: New heuristics and an empirical study. Congressus Numerantium 107, 141–154 (1995)

    MATH  MathSciNet  Google Scholar 

  9. Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximatinng arbitrary metrics by tree metrics. In: Proceedings of STOC 2003, pp. 448–455 (2003)

    Google Scholar 

  10. Galbiati, G.: On min-max cycle bases. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 116–123. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  11. Horton, J.D.: A polynomial-time algorithm to find the shortest cycle basis of a graph. SIAM J. Comput. 16, 358–366 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hu, T.C.: Optimum communication spanning trees. SIAM J. Comput. 3, 188–195 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. Liebchen, C., Peeters, L.: On cyclic timetabling and cycles in graphs. Technical Report 761, Technische Universität Berlin (2002)

    Google Scholar 

  14. Konjevod, G., Ravi, R., Salman, F.S.: On approximating planar metrics by tree metrics. Information Processing Letters 80, 213–219 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  16. Peleg, D.: Polylogarithmic approximation for minimum communication spanning trees. Technical Report CS 97-10, The Weizmann Institute (July 1997)

    Google Scholar 

  17. Peleg, D.: Low stretch spanning trees. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 68–80. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  18. Peleg, D., Reshef, E.: Deterministic polylog approximation for minimum communication spanning trees. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 670–681. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  19. Reshef, E.: Approximating minimum communication cost spanning trees and related problems. M.Sc. Thesis, The Weizmann Institute of Science (1999)

    Google Scholar 

  20. Syslo, M.M.: On cycle bases of a graph. Networks 9, 123–132 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sussenouth Jr., E.: A graph theoretical algorithm for matching chemical structures. J. Chem. Doc. 5, 36–43 (1965)

    Article  Google Scholar 

  22. Vismara, P.: Reconnaissance et représentation d’éléments structuraux pour la description d’objets complexes. Application à l’élaboration de stratégies de synthèse en chimie organique. Thèse de Doctorat, Université de Montpellier II, France (1995)

    Google Scholar 

  23. Wu, B.Y., Lancia, G., Bafna, V., Chao, K.-M., Ravi, R., Tang, C.Y.: A polynomialtime approximation scheme for minimum routing cost spanning trees. SIAM J. Comput. 29, 761–778 (1999)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università degli Studi di Pavia, 27100, Pavia, Italy

    Giulia Galbiati

  2. Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy

    Edoardo Amaldi

Authors
  1. Giulia Galbiati
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  2. Edoardo Amaldi
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Editor information

Editors and Affiliations

  1. Department of Computer Science, The University of Western Ontario, London, Canada

    Roberto Solis-Oba

  2. Institute for Computer Science, University of Kiel, Olshausenstrasse 40, 24118, Kiel, Germany

    Klaus Jansen

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© 2004 Springer-Verlag Berlin Heidelberg

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Galbiati, G., Amaldi, E. (2004). On the Approximability of the Minimum Fundamental Cycle Basis Problem. In: Solis-Oba, R., Jansen, K. (eds) Approximation and Online Algorithms. WAOA 2003. Lecture Notes in Computer Science, vol 2909. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24592-6_12

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  • DOI: https://doi.org/10.1007/978-3-540-24592-6_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21079-5

  • Online ISBN: 978-3-540-24592-6

  • eBook Packages: Springer Book Archive

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