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)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 Ω(1∏j=1log⁎⁡wmax−1log(j)⁡wmax) where wmaxwmax is an upper bound on the edge weights, which may not be known ahead of time. We can improve the competitive ratio to Ω(1log⁡wmax) if wmaxwmax is known to the algorithm when it starts. We also derive an upper bound O(1log⁡wmax) 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 11+2ξ(wmax+1)1ξ, where ξ=min⁡{b,⌈ln⁡(1+wmax)⌉}ξ=min⁡{b,⌈ln⁡(1+wmax)⌉}.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.