Competitive analysis for two variants of online metric matching problem

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