Near optimal algorithms for online maximum edge-weighted b-matching and two-sided vertex-weighted b-matching.
Theoretical Computer Science(2015)
摘要
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 Ω(1∏j=1log⁎wmax−1log(j)wmax) where wmax is an upper bound on the edge weights, which may not be known ahead of time. We can improve the competitive ratio to Ω(1logwmax) if wmax is known to the algorithm when it starts. We also derive an upper bound O(1logwmax) 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)⌉}.
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关键词
Maximum weighted matching,Online algorithms,Competitive analysis,Upper and lower bounds
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