On-line bipartite matching made simple

Abstract

We examine the classic on-line bipartite matching problem studied by Karp, Vazirani, and Vazirani [8] and provide a simple proof of their result that the Ranking algorithm for this problem achieves a competitive ratio of 1 -- 1/ e .

Size versus fairness in the assignment problem

“…They are closely related to the problem of designing an online bilateral matching algorithm maximizing the match size relative to the maximal size feasible o-ine. The Ranking algorithm of [19] selects randomly and uniformly an ordering of the objects, then assigns to the incoming agent the highest acceptable object in that ordering; its m-guaranteed size is no less than 1 (1 1 m+1 ) m (see also [4] for a simpler proof and [18] for a generalization to multiple objects).…”

“…4 Here is an example with …ve agents and four objects. Assume a is the best object for agents 1; 2; 3, b is best for 4; 5, and c; d for nobody.…”

When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

“…The original online stochastic matching problem was studied recently by Feldman et al [10]. They gave a 0.67-competitive algorithm, beating the optimal 1 − 1/e-competitiveness known for worst-case models [17,16,21,5,12]. Our model differs from that in having a bound on the number of items each incoming buyer sees, that each edge is only present with some probability, and that the buyer scans the list linearly (until she times out) and buys the first item she likes.…”

On-Line Maximum Matching in Complete Multipartite Graphs with Implications to the Minimum ADM Problem on a Star Topology

“…Online bipartite maximum matching problem was introduced in [KVV90] and a (1 − 1/e)-competitive randomized algorithm was proposed , which is optimal [GM08,BM08]. The greedy algorithm is 1/2-competitive and is optimal for deterministic online algorithms.…”