Near Optimal Algorithms for Online Maximum Weighted b-Matching

We study the online maximum weighted b- matching problem, in which the input is a 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. Assume that the edge weights in G are no more than w(max), which may not be known ahead of time. We show that a randomized algorithm Greedy-RT which has competitive ratio O(1 log*w(max)-1 j= 1 log(j) wmax). We can improve the competitive ratio to Omega(1 log wmax) if wmax is known to the algorithm when it starts. We also derive an upper bound Omega(1 log wmax) suggesting that Greedy-RT is near optimal. Deterministic algorithms are also considered and we present a near optimal algorithm Greedy-D which is 1/1+2.(w(max)+1) 1/xi-competitive, where xi = min{b, [ln(1 + w(max))]}. We propose a variant of the problem called online two-sided vertex-weighted matching problem, and give a modification of the randomized algorithm Greedy-RT called Greedy-vRT specially for this variant. We show that Greedy-vRT is also near optimal.