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

Tiedemann, et al. [Proc. of WALCOM, LNCS 8973, 2015, pp.210-221] defined multi-objective online problems (as an online version of multi-objective optimization problems) and the competitive analysis for multi-objective online problems and showed that (1) with respect to the worst component competitive analysis, the online algorithm RPP-HIGH is best possible for the multi-objective time series search~problem; (2) with respect to the arithmetic mean component competitive analysis, the online algorithm RPP-MULT is best possible for the bi-objective time series search problem; (3) with respect to the geometric mean component competitive analysis, the online algorithm RPP-MULT is best possible for the bi-objective time series search problem. 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. Then 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 continuous function f). Under the modified framework, we derive exact values of the competitive ratio for the multi-objective time series search problem with respect to the worst component competitive analysis, the arithmetic mean component competitive analysis, and the geometric mean component competitive analysis.