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Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in O(1) Amortized Update Time

  • Sayan BhattacharyaEmail author
  • Deeparnab Chakrabarty
  • Monika Henzinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)

Abstract

We consider the problems of maintaining approximate maximum matching and minimum vertex cover in a dynamic graph. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized 2-approximation dynamic algorithm for this problem that has amortized update time of O(1) with high probability. We consider the natural open question of derandomizing this result. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of \((2+\epsilon )\) and an amortized update time of \(O(\log n/\epsilon ^2)\). Our result can be generalized to give a fully dynamic \(O(f^3)\)-approximation algorithm with \(O(f^2)\) amortized update time for the hypergraph vertex cover and fractional matching problems, where every hyperedge has at most f vertices.

Keywords

Approximation Ratio Vertex Cover Input Graph Maximum Match Dynamic Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abboud, A., Williams, V.V.: Popular conjectures imply strong lower bounds for dynamic problems. In: FOCS (2014)Google Scholar
  2. 2.
    Baswana, S., Gupta, M., Sen, S.: Fully dynamic maximal matching in \(O(\log n)\) update time. In: FOCS (2011)Google Scholar
  3. 3.
    Bernstein, A., Stein, C.: Faster fully dynamic matchings with small approximation ratios. In: SODA (2016)Google Scholar
  4. 4.
    Bhattacharya, S., Henzinger, M., Italiano, G.F.: Design of dynamic algorithms via primal-dual method. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 206–218. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47672-7_17 CrossRefGoogle Scholar
  5. 5.
    Bhattacharya, S., Henzinger, M., Italiano, G.F.: Deterministic fully dynamic data structures for vertex cover and matching. In: SODA (2015)Google Scholar
  6. 6.
    Bhattacharya, S., Henzinger, M., Nanongkai, D.: New deterministic approximation algorithms for fully dynamic matching. In: STOC (2016)Google Scholar
  7. 7.
    Gupta, A., Krishnaswamy, R., Kumar, A., Panigrahi, D.: Online and dynamic algorithms for set cover. In: STOC (2017)Google Scholar
  8. 8.
    Gupta, M., Peng, R.: Fully dynamic \((1+\epsilon )\)-approximate matchings. In: FOCS (2013)Google Scholar
  9. 9.
    Henzinger, M., Krinninger, S., Nanongkai, D., Saranurak, T.: Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In: STOC (2015)Google Scholar
  10. 10.
    Henzinger, M.R., Fredman, M.L.: Lower bounds for fully dynamic connectivity problems in graphs. Algorithmica 22(3), 351–362 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Neiman, O., Solomon, S.: Simple deterministic algorithms for fully dynamic maximal matching. In: STOC (2013)Google Scholar
  12. 12.
    Onak, K., Rubinfeld, R.: Maintaining a large matching and a small vertex cover. In: STOC (2010)Google Scholar
  13. 13.
    Patrascu, M.: Lower bounds for dynamic connectivity. In: Encyclopedia of Algorithms, pp. 1162–1167 (2016)Google Scholar
  14. 14.
    Sankowski, P.: Faster dynamic matchings and vertex connectivity. In: SODA (2007)Google Scholar
  15. 15.
    Solomon, S.: Fully dynamic maximal matching in constant update time. In: FOCS (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sayan Bhattacharya
    • 1
    Email author
  • Deeparnab Chakrabarty
    • 2
  • Monika Henzinger
    • 3
  1. 1.University of WarwickCoventryUK
  2. 2.Department of Computer ScienceDartmouth CollegeHanoverUSA
  3. 3.University of ViennaViennaAustria

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