New Results on Routing via Matchings on Graphs

  • Indranil BanerjeeEmail author
  • Dana Richards
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10472)


In this paper we present some new complexity results on the routing time of a graph under the routing via matching model. This is a parallel routing model which was introduced by Alon et al. [1]. The model can be viewed as a communication scheme on a distributed network. The nodes in the network can communicate via matchings (a step), where a node exchanges data (pebbles) with its matched partner. Let G be a connected graph with vertices labeled from \(\{1,...,n\}\) and the destination vertices of the pebbles are given by a permutation \(\pi \). The problem is to find a minimum step routing scheme for the input permutation \(\pi \). This is denoted as the routing time \(rt(G,\pi )\) of G given \(\pi \). In this paper we characterize the complexity of some known problems under the routing via matching model and discuss their relationship to graph connectivity and clique number. We also introduce some new problems in this domain, which may be of independent interest.


  1. 1.
    Alon, N., Chung, F.R., Graham, R.L.: Routing permutations on graphs via matchings. SIAM J. Discrete Math. 7(3), 513–530 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gyárfás, A., Ruszinkó, M., Sárközy, G., Szemerédi, E.: Partitioning 3-colored complete graphs into three monochromatic cycles. Electronic J. Comb. 18, 1–16 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. J. ACM (JACM) 43(2), 268–292 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kawahara, J., Saitoh, T., Yoshinaka, R.: The time complexity of the token swapping problem and its parallel variants. In: Poon, S.-H., Rahman, Md, Yen, H.-C. (eds.) WALCOM 2017. LNCS, vol. 10167. Springer, Cham (2017). doi: 10.1007/978-3-319-53925-6_35 Google Scholar
  5. 5.
    Zhang, L.: Optimal bounds for matching routing on trees. SIAM J. Discrete Math. 12(1), 64–77 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li, W.T., Lu, L., Yang, Y.: Routing numbers of cycles, complete bipartite graphs, and hypercubes. SIAM J. Discrete Math. 24(4), 1482–1494 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Micali, S., Vazirani, V.V.: An \(O (\sqrt{|V|}|E|)\) algoithm for finding maximum matching in general graphs. In: 1980 21st Annual Symposium on Foundations of Computer Science, pp. 17–27. IEEE, October 1980Google Scholar
  8. 8.
    Yamanaka, K., Demaine, E.D., Ito, T., Kawahara, J., Kiyomi, M., Okamoto, Y., Toshiki, S., Akira, S., Kei, U., Uno, T.: Swapping labeled tokens on graphs. In: Theoretical Computer Science, vol. 586, pp. 81–94 (2015). Computer programming: sorting and searching (Vol. 3). Pearson EducationGoogle Scholar
  9. 9.
    Miltzow, T., Narins, L., Okamoto, Y., Rote, G., Thomas, A., Uno, T.: Approximation and hardness for token swapping. arXiv preprint arXiv:1602.05150 (2016)
  10. 10.
    Hall, M.: Combinatorial Theory, vol. 71. Wiley, Hoboken (1998)zbMATHGoogle Scholar
  11. 11.
    Even, S., Goldreich, O.: The minimum-length generator sequence problem is NP-hard. J. Algorithms 2(3), 311–313 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Feige, U.: Approximating maximum clique by removing subgraphs. SIAM J. Discrete Math. 18(2), 219–225 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Boppana, R., Halldrsson, M.M.: Approximating maximum independent sets by excluding subgraphs. BIT Numer. Math. 32(2), 180–196 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Engebretsen, L., Holmerin, J.: Clique is hard to approximate within n 1-o(1). In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 2–12. Springer, Heidelberg (2000). doi: 10.1007/3-540-45022-X_2 CrossRefGoogle Scholar
  15. 15.
    Jerrum, M.R.: The complexity of finding minimum-length generator sequences. Theor. Comput. Sci. 36, 265–289 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Banerjee, I., Richards, D.: Routing and sorting via matchings on graphs. arXiv preprint arXiv:1604.04978 (2016)
  17. 17.
    Benjamini, I., Shinkar, I., Tsur, G.: Acquaintance time of a graph. SIAM J. Discrete Math. 28(2), 767–785 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Banerjee, I., Richards, D.: New results on routing via matchings on graphs. arXiv preprint arXiv:1706.09355 (2017)

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceGeorge Mason UniversityFairfaxUSA

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