Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line

Author Sharath Raghvendra

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Sharath Raghvendra

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Sharath Raghvendra. Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


In the online metric bipartite matching problem, we are given a set S of server locations in a metric space. Requests arrive one at a time, and on its arrival, we need to immediately and irrevocably match it to a server at a cost which is equal to the distance between these locations. A alpha-competitive algorithm will assign requests to servers so that the total cost is at most alpha times the cost of M_{Opt} where M_{Opt} is the minimum cost matching between S and R. We consider this problem in the adversarial model for the case where S and R are points on a line and |S|=|R|=n. We improve the analysis of the deterministic Robust Matching Algorithm (RM-Algorithm, Nayyar and Raghvendra FOCS'17) from O(log^2 n) to an optimal Theta(log n). Previously, only a randomized algorithm under a weaker oblivious adversary achieved a competitive ratio of O(log n) (Gupta and Lewi, ICALP'12). The well-known Work Function Algorithm (WFA) has a competitive ratio of O(n) and Omega(log n) for this problem. Therefore, WFA cannot achieve an asymptotically better competitive ratio than the RM-Algorithm.
  • Bipartite Matching
  • Online Algorithms
  • Adversarial Model
  • Line Metric


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