Optimal Online Algorithms for the Multi-objective Time Series Search Problem

Abstract

Tiedemann, et al. [Proc. of WALCOM, LNCS 8973, 2015, pp.210-221] defined multiobjective online problems and the competitive analysis for multi-objective online problems, and showed best possible online algorithms with respect to several measures of the competitive analysis. In this paper, we first point out that the definitions and frameworks of the competitive analysis due to Tiedemann, et al. do not necessarily capture the efficiency of online algorithms for multi-objective online problems and provide modified definitions of the competitive analysis for multi-objective online problems. Under the modified framework, we present a simple online algorithm Balanced Price Policy (bpp k ) for the multi-objective (k-objective) time series search problem, and show that the algorithm bpp k is best possible with respect to any measure of the competitive analysis (defined by a monotone function f ). For the modified framework, we also derive best possible values of the competitive ratio for the multi-objective time series search problem with respect to several representative measures of the competitive analysis.

On the Hardness of Deriving the Arithmetic Mean Component Competitive Ratio

“…As we have mentioned, Hasegawa and Itoh [5] derived a closed formula of the arithmetic mean component competitive ratio for k = 2. In this section, we derive a closed formula of the arithmetic mean component competitive ratio for k = 2 given in Eq.…”

“…Without loss of generality, assume that Tiedemann,et al [10] presented best possible online algorithms for the multi-objective time series search problem with respect to the monotone functions f 1 , f 2 , and f 3 , i.e., the best possible online algorithm rpp-high for the multi-objective time series search problem with respect to the monotone function f 1 [10, Theorems 1 and 2], the best possible online algorithm rpp-mult for the bi-objective time series search problem with respect to the monotone function f 2 [10, Theorems 3 and 4] and the best possible online algorithm rpp-mult for the bi-objective time series search problem with respect to the monotone function f 3 [10, §3.2]. Recently, Hasegawa and Itoh [5] presented the deterministic online algorithm balanced price policy bpp and showed that bpp is best possible for any monotone function f : R k → R and for any integer k ≥ 2.…”

“…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, but this does not necessarily imply that it is difficult to write out closed formulas of the arithmetic mean component competitive ratio for a fixed integer k ≥ 2. In fact, Hasegawa and Itoh [5] derived a closed formula of the arithmetic mean component competitive ratio for k = 2. In this paper, we also derive closed formulas of the arithmetic mean component competitive ratio for k = 3 and k = 4.…”

“…and Hasegawa and Itoh [5,Theorem 3.1] showed that with respect to any monotone function f : R k → R, the competitive ratio for the k-objective time series search problem is given by…”

Erratum: Competitive Analysis for Multi-Objective Online Algorithms

“…However, in [2], a best possible algorithm for the bi-objective time series search problem is presented, as outlined below.…”