On the Hardness of Deriving the Arithmetic Mean Component Competitive Ratio

Abstract

For the multi-objective time series search problem, Hasegawa and Itoh [Theoretical Computer Science, Vo.718, pp.58-66, 2018] presented the best possible online algorithm balanced price policy (bpp) 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 closed formula 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 show that it is N P-hard to derive closed formulas of the arithmetic mean component competitive ratio for general integer k ≥ 2. On the the hand, we derive closed formulas of the arithmetic mean component competitive ratio for k = 3 and k = 4.