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

Abstract

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 Dynamic 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 [11,15]. Epstein et al.[5] gave a 5.356-competitive randomized algorithm for MWM, and also proved a lower bound on the competitive ratio of (1 + ln 2) ≈ 1.693 for MCM. The same lower bound applies for MWM.In the Incremental Dynamic Graph model, at each step an edge is added to the graph, and the algorithm is supposed to quickly update its current matching. Gupta [7] proved that for any ǫ ≤ 1/2, there exists an algorithm that maintains a (1+ǫ)-approximate MCM for an incremental bipartite graph in an "amortized" update time of O log 2 n 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 (in expectation) 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 dynamic 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. Hence, we derandomize this algorithm, and give a (3/2 + ǫ)-approximate deterministic algorithm for MCM on trees, with an amortized update time of O(1/ǫ).We also present a minor result for MWM in the online preemptive model, a 3-competitive (in expectation) 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 Matching with General Arrivals

“…The answer is "yes" for fractional algorithms, as shown by combining our Theorem 1.1 with the 0.526-competitive fractional online matching algorithm under general vertex arrivals of Wang and Wong [25]. For integral online matching, however, the problem has proven challenging, and the only positive results for this problem, too, are for various relaxations, such as restriction to trees, either with or without preemption [3,4,24], for bounded-degree graphs [3], or (recently) allowing vertices to be matched during some known time interval [15,16].…”

“…(Both bounds apply even to online algorithms with preemption; i.e., allowing edges to be removed from the matching in favor of a newly-arrived edge.) On the positive side, as pointed out by Buchbinder et al [3], the edge arrival model has proven challenging, and results beating the 1 /2 competitive ratio were only achieved under various relaxations, including: random order edge arrival [14], bounded number of arrival batches [20], on trees, either with or without preemption [3,24], and for bounded-degree graphs [3]. The above papers all asked whether there exists a randomized ( 1 /2 + Ω(1))-competitive algorithm for adversarial edge arrivals (see also Open Question 17 in Mehta's survey [22]).…”