Relaxing the Irrevocability Requirement for Online Graph Algorithms

  • Joan Boyar
  • Lene M. Favrholdt
  • Michal Kotrbčík
  • Kim S. Larsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is \(\frac{3}{2}\).

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References

  1. 1.
    Bartal, Y., Fiat, A., Leonardi, S.: Lower bounds for on-line graph problems with application to on-line circuit and optical routing. In: 28th STOC, pp. 531–540. ACM (1996)Google Scholar
  2. 2.
    Boyar, J., Eidenbenz, S.J., Favrholdt, L.M., Kotrbčík, M., Larsen, K.S.: Online dominating set. In: 15th SWAT, LIPIcs, vol. 53, pp. 21:1–21:15. Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH (2016)Google Scholar
  3. 3.
    Boyar, J., Favrholdt, L.M., Kotrbčík, M., Larsen, K.S.: Relaxing the irrevocability requirements for online graph algorithms. Technical Report arXiv:1704.08835 [cs.DS], arXiv (2017)
  4. 4.
    Cygan, M., Jeż, Ł., Sgall, J.: Online knapsack revisited. Theor. Comput. Syst. 58(1), 153–190 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Demange, M., Paschos, V.T.: On-line vertex-covering. Theor. Comput. Sci. 332, 83–108 (2005)Google Scholar
  6. 6.
    Epstein, L., Levin, A., Mestre, J., Segev, D.: Improved approximation guarantees for weighted matching in the semi-streaming model. SIAM J. Discrete Math. 25(3), 1251–1265 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Epstein, L., Levin, A., Segev, D., Weimann, O.: Improved bounds for online preemptive matching. In: 30th STACS, LIPIcs, vol. 20, pp. 389–399. Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH (2013)Google Scholar
  8. 8.
    Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: On graph problems in a semi-streaming model. Theor. Comput. Sci. 348(2–3), 207–216 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Garay, J.A., Gopal, I.S., Kutten, S., Mansour, Y., Yung, M.: Efficient on-line call control algorithms. J. Algorithm. 23(1), 180–194 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gu, A., Gupta, A., Kumar, A.: The power of deferral: Maintaining a constant-competitive steiner tree online. SIAM J. Comput. 45(1), 1–28 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gupta, A., Kumar, A.: Online steiner tree with deletions. In: 25th SODA, pp. 455–467 (2014)Google Scholar
  12. 12.
    Han, X., Kawase, Y., Makino, K.: Randomized algorithms for online knapsack problems. Theor. Comput. Sci. 562, 395–405 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Han, X., Kawase, Y., Makino, K., Guo, H.: Online removable knapsack problem under convex function. Theor. Comput. Sci. 540, 62–69 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Han, X., Makino, K.: Online minimization knapsack problem. Theor. Comput. Sci. 609, 185–196 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Imase, M., Waxman, B.M.: Dynamic steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Iwama, K., Taketomi, S.: Removable online knapsack problems. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 293–305. Springer, Heidelberg (2002). doi:10.1007/3-540-45465-9_26 CrossRefGoogle Scholar
  17. 17.
    Jaillet, P., Lu, X.: Online traveling salesman problems with rejection options. Networks 64, 84–95 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3, 79–119 (1988)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Komm, D., Královič, R., Královič, R., Kudahl, C.: Advice complexity of the online induced subgraph problem. In: 41st MFCS, LIPIcs, vol. 58, pp. 59:1–59:13. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)Google Scholar
  20. 20.
    Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Ann. Discrete Math. 2, 65–74 (1978)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Megow, N., Skutella, M., Verschae, J., Wiese, A.: The power of recourse for online MST and TSP. SIAM J. Comput. 45(3), 859–880 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rawitz, D., Rosén, A.: Online budgeted maximum coverage. In: 24th ESA, LIIPCcs, vol. 57, pp. 73:1–73:17. Schloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH (2016)Google Scholar
  23. 23.
    Saha, B., Getoor, L.: On maximum coverage in the streaming model & application to multi-topic blog-watch. In: 9th SDM, pp. 697–708. SIAM (2009)Google Scholar
  24. 24.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Communications of the ACM 28(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tarjan, R.E.: Data structures and network algorithms. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 44. SIAM (1983)Google Scholar
  26. 26.
    Vinkemeier, D.E.D., Hougardy, S.: A linear-time approximation algorithm for weighted matchings in graphs. ACM T. Algorithms 1(1), 107–122 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Joan Boyar
    • 1
  • Lene M. Favrholdt
    • 1
  • Michal Kotrbčík
    • 2
  • Kim S. Larsen
    • 1
  1. 1.University of Southern DenmarkOdenseDenmark
  2. 2.The University of QueenslandBrisbaneAustralia

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