宮崎 修一

In the online facility assignment problem [Formula: see text], there exist [Formula: see text] servers [Formula: see text] on a metric space where each [Formula: see text] has an integer capacity [Formula: see text] 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 [Formula: see text], we consider [Formula: see text] on a line , which is denoted by [Formula: see text] and [Formula: see text], where the latter is the case of [Formula: see text] with equidistant servers. In this paper, we perform 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 MPFS algorithm alg for [Formula: see text], if alg is [Formula: see text]-competitive when [Formula: see text], then alg is [Formula: see text]-competitive for general [Formula: see text]. By applying the capacity-insensitive property of the greedy algorithm grdy, we derive the matching upper and lower bounds [Formula: see text] on the competitive ratio of grdy for [Formula: see text]. To investigate the capability of MPFS algorithms, we show that the competitive ratio of any MPFS algorithm alg for [Formula: see text] is at least [Formula: see text]. Then, we propose a new MPFS algorithm idas (Interior Division for Adjacent Servers) for [Formula: see text] and show that the competitive ratio of idas for [Formula: see text] is at most [Formula: see text], i.e., idas for [Formula: see text] is best possible in all the MPFS algorithms. We also give numerical experiments to investigate the performance of idas and grdy and show that idas performs better than grdy for distribution of request sequences with locality.

This paper has two objectives. One is to give a linear time algorithm that solves the stable roommates problem (i.e., obtains one stable matching) using the stable marriage problem. The idea is that a stable matching of a roommate instance [Formula: see text] is a stable matching (that however must satisfy a certain condition) of some marriage instance [Formula: see text]. [Formula: see text] is obtained just by making two copies of [Formula: see text], one for the men’s table and the other for the women’s table. The second objective is to investigate the possibility of reducing the roommate problem to the marriage problem (with one-to-one correspondence between their stable matchings) in polynomial time. For a given [Formula: see text], we construct the rotation POSET [Formula: see text] of [Formula: see text] and then we “halve” it to obtain [Formula: see text], by which we can forget the above condition and can use all the closed subsets of [Formula: see text] for all the stable matchings of [Formula: see text]. Unfortunately this approach works (runs in polynomial time) only for restricted instances.

In the online metric matching problem, there are servers on a given metric space and requests are given one-by-one. The task of an online algorithm is to match each request immediately and irrevocably with one of the unused servers. In this paper, we pursue competitive analysis for two variants of the online metric matching problem. The first variant is a restriction where each server is placed at one of two positions, which is denoted by OMM([Formula: see text]). We show that a simple greedy algorithm achieves the competitive ratio of 3 for OMM([Formula: see text]). We also show that this greedy algorithm is optimal by showing that the competitive ratio of any deterministic online algorithm for OMM([Formula: see text]) is at least 3. The second variant is the online facility assignment problem on a line. In this problem, the metric space is a line, the servers have capacities, and the distances between any two consecutive servers are the same. We denote this problem by OFAL([Formula: see text]), where [Formula: see text] is the number of servers. We first observe that the upper and lower bounds for OMM([Formula: see text]) also hold for OFAL([Formula: see text]), so the competitive ratio for OFAL([Formula: see text]) is exactly 3. We then show lower bounds on the competitive ratio [Formula: see text] [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] for OFAL([Formula: see text]), OFAL([Formula: see text]) and OFAL([Formula: see text]), respectively.