On the Hardness of Deriving the Arithmetic Mean Component Competitive Ratio

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 short) for any monotone function $f: R^k \to R$. Specifically, the competitive ratio with respect to the monotone function $f(c_{1},\ldots,c_{k})=(c_{1}+\cdots+c_{k})/k$ 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 $k \geq 3$. In this paper, we show that it is NP-hard to derive closed formulas of the arithmetic mean component competitive ratio for general integer $k\geq 2$. On the the hand, we derive closed formulas of the arithmetic mean component competitive ratio for $k=3$ and $k=4$.