On the Competitive Analysis for the Multi-Objective Time Series Search Problem

For the multi-objective time series search problem, Hasegawa and Itoh [Theoretical Computer Science, Vol. 78, pp.58-66, 2018] presented the best possible online algorithm balanced price policy for any monotone function f : R-k -> R. Specifically the competitive ratio with respect to the monotone function f (c(1), . . . , c(k)) = (c(1) + . . . +c(k))/k is referred to as the arithmetic mean component competitive ratio. Hasegawa and Itoh derived the explicit representation of the arithmetic mean component competitive ratio for k = 2, but it has not been known for any integer k >= 3. In this paper, we derive the explicit representations of the arithmetic mean component competitive ratio for k = 3 and k = 4, respectively. On the other hand, we show that it is computationally difficult to derive the explicit representation of the arithmetic mean component competitive ratio for arbitrary integer k in a way similar to the cases for k = 2, 3, and 4.