Near optimal algorithms for online maximum edge-weighted b-matching and two-sided vertex-weighted b-matching

This paper studies the online maximum edge-weighted b-matching problem. The input of the problem is a weighted bipartite graph G = (L, R, E, w). Vertices in R arrive online, and each vertex in L can be matched to at most b vertices in R. The objective is to maximize the total weight of the matching edges. We give a randomized algorithm GREEDY-RT for this problem, and show that its competitive ratio is Omega(1/Pi(log*wmax-1)(J=1) log((j)) w(max)) where w(max) is an upper bound on the edge weights, which may not be known ahead of time. We can improve the competitive ratio to Omega(1/logw(max)) if w(max) is known to the algorithm when it starts. We also derive an upper bound O(1/logw(max)) on the competitive ratio, suggesting that GREEDY-RT is near optimal. We also consider deterministic algorithms; we present a near optimal algorithm GREEDY-D which has competitive ratio 1/1+2 xi(w(max)+1)(1/xi) where xi = min{b, left perpendicular ln(1 + w(max))right perpendicular}. We also study a variant of the problem called online maximum two-sided vertex-weighted b-matching problem, and give a modification of the randomized algorithm GREEDY-RT called GREEDY-vRT for this variant. We show that the competitive ratio of GREEDY-vRT is also near optimal. (C) 2015 Elsevier B.V. All rights reserved.