In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is 1/2-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs. We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a 5/9-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of 2 3+1/φ2≈0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, this bound holds even for an easier model in which vertices (along with their adjacent edges) arrive online, and when the underlying graph is a tree of maximum degree at most 3.
|Title of host publication||25th European Symposium on Algorithms, ESA 2017|
|Editors||Christian Sohler, Christian Sohler, Kirk Pruhs|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Sep 2017|
|Event||25th European Symposium on Algorithms, ESA 2017 - Vienna, Austria|
Duration: 4 Sep 2017 → 6 Sep 2017
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||25th European Symposium on Algorithms, ESA 2017|
|Period||4/09/17 → 6/09/17|
Bibliographical noteFunding Information:
∗ The research of Niv Buchbinder is supported by ISF grant 1585/15 and US-Israel BSF grant 2014414. The research of Danny Segev is supported by ISF grant 148/16.
© Niv Buchbinder, Danny Segev, and Yevgeny Tkach.
- Competitive analysis
- Maximum matching
- Online algorithms
- Primal-dual method
ASJC Scopus subject areas