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Finding Two Edge-Disjoint Paths with Length Constraints

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Graph-Theoretic Concepts in Computer Science (WG 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9941))

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Abstract

We consider the problem of finding, for two pairs \((s_1,t_1)\) and \((s_2,t_2)\) of vertices in an undirected graph, an \((s_1,t_1)\)-path \(P_1\) and an \((s_2,t_2)\)-path \(P_2\) such that \(P_1\) and \(P_2\) share no edges and the length of each \(P_i\) satisfies \(L_i\), where \(L_i \in \{ \le k_i, \; = k_i, \; \ge k_i, \; \le \infty \}\). We regard \(k_1\) and \(k_2\) as parameters and investigate the parameterized complexity of the above problem when at least one of \(P_1\) and \(P_2\) has a length constraint (note that \(L_i = ``\le \infty \)” indicates that \(P_i\) has no length constraint). For the nine different cases of \((L_1, L_2)\), we obtain FPT algorithms for seven of them. Our algorithms uses random partition backed by some structural results. On the other hand, we prove that the problem admits no polynomial kernel for all nine cases unless \(NP \subseteq coNP/poly\).

Partially supported by GRF grant CUHK410212 of the Research Grants Council of Hong Kong.

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References

  1. Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Jansen, B.M., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28(1), 277–305 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci. 412, 4570–4578 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, L., Cai, Y.: Incompressibility of H-free edge modification problems. Algorithmica 71(3), 731–757 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M., Schlotter, I.: Parameterized complexity of Eulerian deletion problems. Algorithmica 68(1), 41–61 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Eilam-Tzoreff, T.: The disjoint shortest paths problem. Discrete Appl. Math. 85(2), 113–138 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Representative sets of product families. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 443–454. Springer, Heidelberg (2014)

    Google Scholar 

  10. Fomin, F.V., Lokshtanov, D., Saurabh, S.: Efficient computation of representative sets with applications in parameterized and exact algorithms. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 142–151. SIAM (2014)

    Google Scholar 

  11. Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gabow, H.N., Nie, S.: Finding long paths, cycles and circuits. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 752–763. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Golovach, P.A., Thilikos, D.M.: Paths of bounded length and their cuts: parameterized complexity and algorithms. Discrete Optim. 8(1), 72–86 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pp. 182–191. IEEE (1995)

    Google Scholar 

  15. Ohtsuki, T.: The two disjoint path problem and wire routing design. In: Saito, N., Nishizeki, T. (eds.) Graph Theory and Algorithms. LNCS, vol. 108, pp. 207–216. Springer, Heidelberg (1980)

    Chapter  Google Scholar 

  16. Orlin, J.B.: Max flows in O(nm) time, or better. In: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, pp. 765–774. ACM (2013)

    Google Scholar 

  17. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Seymour, P.D.: Disjoint paths in graphs. Discrete Math. 29(3), 293–309 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shachnai, H., Zehavi, M.: Representative families: a unified tradeoff-based approach. J. Comput. Syst. Sci. 82(3), 488–502 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. ACM 27(3), 445–456 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  21. Thomassen, C.: 2-linked graphs. Eur. J. Comb. 1(4), 371–378 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tragoudas, S., Varol, Y.L.: Computing disjoint paths with length constraints. In: d’Amore, F., Franciosa, P.G., Marchetti-Spaccamela, A. (eds.) Graph-Theoretic Concepts in Computer Science. LNCS, vol. 1197, pp. 375–389. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  23. Zehavi, M.: Mixing color coding-related techniques. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 1037–1049. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China

    Leizhen Cai & Junjie Ye

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  1. Leizhen Cai
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  2. Junjie Ye
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Correspondence to Junjie Ye .

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  1. University of Bergen, Bergen, Norway

    Pinar Heggernes

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Cai, L., Ye, J. (2016). Finding Two Edge-Disjoint Paths with Length Constraints. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_6

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  • DOI: https://doi.org/10.1007/978-3-662-53536-3_6

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