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Maximum Matching on Trees in the Online Preemptive and the Incremental Graph Models

  • Sumedh TirodkarEmail author
  • Sundar Vishwanathan
Part of the following topical collections:
  1. Special Issue: Computing and Combinatorics


We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the online preemptive and the incremental graph models. In the Online Preemptive model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms (McGregor, in: Proceedings of the 8th international workshop on approximation, randomization and combinatorial optimization problems, and proceedings of the 9th international conference on randomization and computation: algorithms and techniques, APPROX’05/RANDOM’05, Springer, Berlin, pp. 170–181, 2005; Varadaraja, in: Automata, languages and programming: 38th international colloquium, ICALP 2011, Zurich, Switzerland, proceedings, part I, pp. 379–390, 2011. Epstein et al. (in: 30th international symposium on theoretical aspects of computer science, STACS 2013, Kiel, Germany, pp. 389–399, 2013. gave a 5.356-competitive randomized algorithm for MWM, and also proved a lower bound on the competitive ratio of \((1+\ln 2) \approx 1.693\) for MCM. The same lower bound applies for MWM. In the Incremental Graph model, at each step an edge is added to the graph, and the algorithm is supposed to quickly update its current matching. Gupta (in: 34th international conference on foundation of software technology and theoretical computer science, FSTTCS 2014, 15–17 Dec 2014, New Delhi, India, pp. 227–239, 2014. proved that for any \(\epsilon \le 1/2\), there exists an algorithm that maintains a \((1+\epsilon )\)-approximate MCM for an incremental bipartite graph in an amortized update time of \(O\left( \frac{\log ^2 n}{\epsilon ^4}\right) \). No \((2-\epsilon )\)-approximation algorithm with a worst case update time of O(1) is known in this model, even for special classes of graphs. In this paper we show that some of the results can be improved for trees, and for some special classes of graphs. In the online preemptive model, we present a 64 / 33-competitive randomized algorithm (which uses only two bits of randomness) for MCM on trees. Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a worst case update time of O(1), in the incremental graph model, which is 3 / 2-approximate (in expectation) on trees, and 1.8-approximate (in expectation) on general graphs with maximum degree 3. Note that this algorithm works only against an oblivious adversary. We derandomize this algorithm, and give a \((3/2+\epsilon )\)-approximate deterministic algorithm for MCM on trees, with an amortized update time of \(O(1/\epsilon )\). We also present a minor result for MWM in the online preemptive model, a 3-competitive randomized algorithm (that uses only O(1) bits of randomness) on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).


Online preemptive model Incremental dynamic graph model Primal-dual analysis 



The first author would like to thank Ashish Chiplunkar for helpful suggestions to improve the competitive ratio of Algorithm 4, and also to improve the presentation of Sect. 4.


  1. 1.
    Bernstein, A., Stein, C.: Faster fully dynamic matchings with small approximation ratios. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, 10–12 Jan 2016, pp. 692–711. (2016)
  2. 2.
    Bhattacharya, S., Henzinger, M., Nanongkai, D.: Fully dynamic approximate maximum matching and minimum vertex cover in O(log\({}^{\text{3}}\) n) worst case update time. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, 16–19 Jan 2017, pp. 470–489. (2017)
  3. 3.
    Chiplunkar, A., Tirodkar, S., Vishwanathan, S.: On randomized algorithms for matching in the online preemptive model. In: Algorithms-ESA 2015—23rd Annual European Symposium, Patras, Greece, 14–16 Sept 2015, Proceedings, pp. 325–336. (2015)
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Epstein, L., Levin, A., Segev, D., Weimann, O.: Improved bounds for online preemptive matching. In: 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, Kiel, Germany, pp. 389–399. (2013)
  6. 6.
    Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: On graph problems in a semi-streaming model. Theor. Comput. Sci. 348(2), 207–216 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gupta, M.: Maintaining approximate maximum matching in an incremental bipartite graph in polylogarithmic update time. In: 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2014, 15–17 Dec 2014, New Delhi, India, pp. 227–239. (2014)
  8. 8.
    Gupta, M., Peng, R.: Fully dynamic \((1+ \epsilon )\)-approximate matchings. In: Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS ’13, pp. 548–557, Washington, DC, USA, 2013. IEEE Computer Society (2013).
  9. 9.
    Kalyanasundaram, B., Pruhs, K.: Online weighted matching. J. Algorithms 14(3), 478–488 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, STOC ’90, pp. 352–358, New York, NY, USA. ACM (1990)Google Scholar
  11. 11.
    McGregor, A.: Finding graph matchings in data streams. In: Proceedings of the 8th International Workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th International Conference on Randomization and Computation: Algorithms and Techniques, APPROX’05/RANDOM’05, pp. 170–181. Springer, Berlin (2005)Google Scholar
  12. 12.
    Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, SFCS ’80, pp. 17–27, Washington, DC, USA. IEEE Computer Society (1980).
  13. 13.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)zbMATHGoogle Scholar
  14. 14.
    Solomon, S.: Fully dynamic maximal matching in constant update time. In: IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, Oct 9–11 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pp. 325–334. (2016)
  15. 15.
    Varadaraja, A.B.: Buyback problem: approximate matroid intersection with cancellation costs. In: Automata, Languages and Programming: 38th International Colloquium, ICALP 2011, Zurich, Switzerland, Proceedings, Part I, pp. 379–390. (2011)

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Authors and Affiliations

  1. 1.Tata Institute of Fundamental ResearchMumbaiIndia
  2. 2.CSE, IIT BombayMumbaiIndia

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