Capacity-Insensitive Algorithms for Online Facility Assignment Problems on a Line
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In the online facility assignment problem OFA(k,ℓ), there exist k servers with a capacity ℓ≥1 on a metric space and a request arrives one-by-one. The task of an online algorithm is to irrevocably match a current request with one of the servers with vacancies before the next request arrives. As special cases for OFA(k,ℓ), we consider OFA(k,ℓ) on a line, which is denoted by OFAL(k,ℓ) and OFAL_eq(k,ℓ), where the latter is the case of OFAL(k,ℓ) with equidistant servers. In this paper, we deal with the competitive analysis for the above problems. As a natural generalization of the greedy algorithm GRDY, we introduce a class of algorithms called MPFS (most preferred free servers) and show that any MPFS algorithm has the capacity-insensitive property, i.e., for any ℓ≥1, ALG is c-competitive for OFA(k,1) iff ALG is c-competitive for OFA(k,ℓ). By applying the capacity-insensitive property of the greedy algorithm GRDY, we derive the matching upper and lower bounds 4k-5 on the competitive ratio of GRDY for OFAL_eq(k,ℓ). To investigate the capability of MPFS algorithms, we show that the competitive ratio of any MPFS algorithm ALG for OFAL_eq(k,ℓ) is at least 2k-1. Then we propose a new MPFS algorithm IDAS (Interior Division for Adjacent Servers) for OFAL(k,ℓ) and show that the competitive ratio of IDAS for OFAL_eq(k,ℓ) is at most 2k-1, i.e., IDAS for OFAL_eq(k,ℓ) is best possible in all the MPFS algorithms.
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