Capacity-Insensitive Algorithms for Online Facility Assignment Problems on a Line

In the online facility assignment problem OFA(k,\ell), there exist k servers with a capacity \ell \geq 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,\ell), we consider OFA(k,\ell) on a line, which is denoted by OFAL(k,\ell) and OFAL_{eq}(k,\ell), where the latter is the case of OFAL(k,\ell) 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 \ell \geq 1, ALG is c-competitive for OFA(k,1) iff ALG is c-competitive for OFA(k,\ell). 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,\ell). To investigate the capability of MPFS algorithms, we show that the competitive ratio of any MPFS algorithm ALG for OFAL_{eq}(k,\ell) is at least $2k-1$. Then we propose a new MPFS algorithm IDAS (Interior Division for Adjacent Servers) for OFAL(k,\ell) and show that the competitive ratio of IDAS for OFAL}_{eq}(k,\ell) is at most 2k-1, i.e., IDAS for OFAL_{eq}(k,\ell) is best possible in all the MPFS algorithms.