Near optimal algorithms for online maximum weighted b-matching

Abstract

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 wmax, which may not be known ahead of time. We show that a randomized algorithm Greedy-RT which has competitive ratio Ω(1/Πj=1log* wmax-1 log (j) wmax. We can improve the competitive ratio to Ω(1/log wmax) if wmax is known to the algorithm when it starts. We also derive an upper bound O(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+2ℰ(w max+1)1/ℰ-competitive, where ℰ = 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- v RT specially for this variant. We show that Greedy- v RT is also near optimal. © 2014 Springer International Publishing.