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A \(\frac{5}{4}\)-Approximation for Subcubic 2EC Using Circulations

  • Sylvia Boyd
  • Yao Fu
  • Yu Sun
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

In this paper we study the NP-hard problem of finding a minimum size 2-edge-connected spanning subgraph (henceforth 2EC) in cubic and subcubic multigraphs. We present a new \(\frac{5}{4}\)-approximation algorithm for 2EC for subcubic bridgeless graphs, improving upon the current best approximation ratio of \(\frac{5}{4}+\varepsilon\). Our algorithm involves an elegant new method based on circulations which we feel has potential to be more broadly applied. We also study the closely related integrality gap problem, i.e. the worst case ratio between the integer linear program for 2EC and its linear programming relaxation, both theoretically and computationally. We show this gap is at most \(\frac{9}{8}\) for all subcubic bridgeless graphs with up to 16 nodes. Moreover, we present a family of graphs that demonstrate the integrality gap is at least \(\frac{8}{7}\), even when restricted to subcubic bridgeless graphs. This represents an improvement over the previous best known bound of \(\frac{9}{8}\).

Keywords

minimum 2-edge-connected subgraph problem approximation algorithm circulations integrality gap subcubic graphs 

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References

  1. 1.
    Csaba, B., Karpinski, M., Krysta, P.: Approximability of dense and sparse instances of minimum 2-connectivity, tsp and path problems. In: Eppstein, D. (ed.) SODA, ACM/SIAM, pp. 74–83 (2002)Google Scholar
  2. 2.
    Alexander, A., Boyd, S., Elliott-Magwood, P.: On the integrality gap of the 2-edge connected subgraph problem. Technical Report TR-2006-04, SITE, University of Ottawa, Ottawa, Canada (2006)Google Scholar
  3. 3.
    Mömke, T., Svensson, O.: Approximating graphic tsp by matchings. In: Ostrovsky, R. (ed.) IEEE FOCS, pp. 560–569 (2011)Google Scholar
  4. 4.
    Sebő, A., Vygen, J.: Shorter tours by nicer ears. CoRR abs/1201.1870 (2012)Google Scholar
  5. 5.
    Khuller, S., Vishkin, U.: Biconnectivity approximations and graph carvings. J. ACM 41(2), 214–235 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cheriyan, J., Sebő, A., Szigeti, Z.: Improving on the 1.5 approximation of a smallest 2-edge connected spanning subgraph. SIAM J. Discrete Math. 14, 170–180 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Vempala, S., Vetta, A.: Factor 4/3 approximations for minimum 2-connected subgraphs. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 262–273. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Krysta, P., Kumar, V.S.A.: Approximation algorithms for minimum size 2-connectivity problems. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 431–442. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Huh, W.T.: Finding 2-edge connected spanning subgraphs. Oper. Res. Lett. 32(3), 212–216 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boyd, S., Iwata, S., Takazawa, K.: Finding 2-factors closer to tsp tours in cubic graphs. SIAM J. Discrete Math. 27(2), 918–939 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hoffman, A.J.: Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In: Combinatorial Analysis, pp. 113–127 (1960)Google Scholar
  12. 12.
    Schrijver, A.: Chapters 11-12. In: Combinatorial Optimization. Springer (2003)Google Scholar
  13. 13.
    Sun, Y.: Theoretical and experimental studies on the minimum size 2-edge-connected spanning subgraph problem. Master’s thesis, University of Ottawa, Ottawa, Canada (2013)Google Scholar
  14. 14.
    McKay, B.D.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Sylvia Boyd
    • 1
  • Yao Fu
    • 1
  • Yu Sun
    • 1
  1. 1.School of Electric Engineering and Computer Science (EECS)University of OttawaOttawaCanada

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