Abstract
We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the graph and each vertex must be matched to a right side vertex immediately upon arrival. A vertex cannot be rematched to another vertex once it is matched. The goal is to minimize the total weight of the matching.
We give a O(log2 k) competitive randomized algorithm for the problem. This improves upon the best known guarantee of O(log3 k) due to Meyerson, Nanavati and Poplawski [19] . It is well known that no deterministic algorithm can have a competitive less than 2k − 1, and that no randomized algorithm can have a competitive ratio of less than ln k.
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Bansal, N., Buchbinder, N., Gupta, A., Naor, J.(. (2007). An O(log2 k)-Competitive Algorithm for Metric Bipartite Matching. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_47
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