On Randomized Algorithms for Matching in the Online Preemptive Model

Abstract

We investigate the power of randomized algorithms for the maximum cardinality matching (MCM) and the maximum weight matching (MWM) problems in the online preemptive model. In this 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. The complexity of the problem is settled for deterministic algorithms [6, 8].Almost nothing is known for randomized algorithms. A lower bound of 1.693 is known for MCM with a trivial upper bound of two. An upper bound of 5.356 is known for MWM. We initiate a systematic study of the same in this paper with an aim to isolate and understand the difficulty. We begin with a primal-dual analysis of the deterministic algorithm due to [6]. All deterministic lower bounds are on instances which are trees at every step. For this class of (unweighted) graphs we present a randomized algorithm which is 28 15 -competitive. The analysis is a considerable extension of the (simple) primal-dual analysis for the deterministic case. The key new technique is that the distribution of primal charge to dual variables depends on the "neighborhood" and needs to be done after having seen the entire input. The assignment is asymmetric: in that edges may assign different charges to the two end-points. Also the proof depends on a non-trivial structural statement on the performance of the algorithm on the input tree.The other main result of this paper is an extension of the deterministic lower bound of Varadaraja [8] to a natural class of randomized algorithms which decide whether to accept a new edge or not using independent random choices. This indicates that randomized algorithms will have to use dependent coin tosses to succeed. Indeed, the few known randomized algorithms, even in very restricted models follow this.We also present the best possible 4 3 -competitive randomized algorithm for MCM on paths.

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

“…We use the primal-dual technique to analyze the performance of this algorithm. The primaldual technique used to analyze McGregor's deterministic algorithm for MWM described in [3] is fairly straightforward. However the management becomes complicated with the introduction of randomness, and we are only able to analyze the algorithm in a very restricted class of graphs, which are growing trees.…”

“…Step (3c) in Algorithm 3 ensures that at each stage |M c | ≥ (|M i | + |M j |) / (2(1 + ǫ)), such that i, j, c ∈ [3], and are all distinct, and M c is the current matching which will be output by the algorithm on query. Let M be the optimum matching at any stage.…”

Online maximum matching with recourse

“…The same paper [6] shows that the randomized competitive ratio is between 1 + log 2 ≈ 1.693 and 5.356. Chiplunkar et al [5] presented a randomized 28/15-competitive algorithm for trees and a 4/3-competitive algorithm for paths.…”

“…To quantify the impact of recourse, several models have been proposed that relax the irrevocable nature of a decision. In the late reject model [3], which is also called the preemptive model [5], an edge can be accepted only upon its arrival, but can be later rejected. In the edge-bounded recourse model, introduced in [1], the algorithm can switch between accepting and rejecting an edge that has already appeared, but is allowed up to k such modifications per edge.…”

Deferred On-Line Bipartite Matching

“…Here, edges with weights are incoming on-line and algorithm is allowed to remove previously accepted edges in order to add a new one. A collection of results on preemptive matching can be found in [5,7].…”