Abstract
Tiedemann, et al. [Proc. of WALCOM, LNCS 8973, 2015, pp. 210–221] defined multi-objective online problems and the competitive analysis for multi-objective online problems, and presented best possible online algorithms for the multi-objective online problems with respect to several measures of competitive analysis. In this paper, we first point out that the 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 time series search problem, and show that the algorithm bpp\(_{k}\) is best possible with respect to any measure of the competitive analysis. For 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.
T. Itoh—The author gratefully acknowledges the ELC project (Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan) for encouraging the research presented in this paper.
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Notes
- 1.
Tiedemann, et al. [9] introduced notions of (strong) \({{\varvec{c}}}\)-competitive and (strong) competitive ratio. In this paper, we do not deal with the notion of \({{\varvec{c}}}\)-competitive and competitive ratio. Thus for simplicity, we refer to strong \({{\varvec{c}}}\)-competitive and strong competitive ratio as \({{\varvec{c}}}\)-competitive and competitive ratio, respectively.
- 2.
It is possible to show that if only the fluctuation ratio \(\phi =M/m\) is known (but not m or M) to alg, then no better competitive ratio than the trivial one of \(\phi \) is achievable.
- 3.
If \(\mathcal{J} = \emptyset \), then \(M_{i}/p_{t}^{i} > p_{t}^{i}/m_{i}\) for each \(i \in [1,k]\). Since \(f: \mathbf{R}^{k}\rightarrow \mathbf{R}\) is a monotone function, we have that \(f(M_{1}/p_{t}^{1},\ldots ,M_{k}/p_{t}^{k}) \ge f(p_{t}^{1}/m_{1},\ldots ,p_{t}^{k}/m_{k})\), which contradicts the assumption that \(f(M_{1}/p_{t}^{1},\ldots ,M_{k}/p_{t}^{k}) <f(p_{t}^{1}/m_{1},\ldots ,p_{t}^{k}/m_{k})\).
- 4.
If \(\mathcal{H}=\emptyset \), then \(M_{i}/p_{\tau }^{i} < p_{\tau }^{i}/m_{i}\) for each \(i \in [1,k]\). Since \(f: \mathbf{R}^{k}\rightarrow \mathbf{R}\) is a monotone function, we have that \(f(M_{1}/p_{t}^{1},\ldots ,M_{k}/p_{t}^{k}) \le f(p_{t}^{1}/m_{1},\ldots ,p_{t}^{k}/m_{k})\), which contradicts the assumption that \(f(M_{1}/p_{t}^{1},\ldots , M_{k}/p_{t}^{k})>f(p_{t}^{1}/m_{1},\ldots ,p_{t}^{k}/m_{k})\).
References
Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)
Ehrgott, M.: Multicriteria Optimization. Springer, Heidelberg (2005)
El-Yaniv, R., Fiat, A., Karp, R.M., Turpin, G.: Optimal search and one-way trading online algorithms. Algorithmica 30(1), 101–139 (2001)
Goldwasser, M.H.: A survey of buffer management policies for packet switches. ACM SIGACT New 41(1), 100–128 (2010)
Hasegawa, S., Itoh, T.: Optimal Online Algorithms for the Multi-Objective Time Series Search Problem. CoRR abs/1506.04474 (2015)
Koutsoupias, E.: The \(k\)-server conjecture. Comput. Sci. Rev. 3(2), 105–118 (2009)
Mohr, E., Ahmad, I., Schmidt, G.: Online algorithms for conversion problems: a survey. Surv. Oper. Res. Manag. Sci. 19(2), 87–104 (2014)
Sleator, D.D., Tarjan, R.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1085)
Tiedemann, M., Ide, J., Schöbel, A.: Competitive analysis for multi-objective online algorithms. In: Rahman, M.S., Tomita, E. (eds.) WALCOM 2015. LNCS, vol. 8973, pp. 210–221. Springer, Heidelberg (2015)
Young, N.E.: Online paging and caching. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, pp. 601–604. Springer, Heidelberg (2008)
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Hasegawa, S., Itoh, T. (2016). Optimal Online Algorithms for the Multi-objective Time Series Search Problem. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_24
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