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Finite-State Online Algorithms and Their Automated Competitive Analysis

By Takashi Horiyama, Kazuo Iwama and Jun Kawahara


Abstract. In this paper we study the Revocable Online Knapsack Problem (ROKP) which is an extension of the Online Knapsack Problem [8]. We prove an optimal upper bound of 1/t for the competitive ratio of ROKP, where t is a real root of 4x 3 + 5x 2 − x − 4 = 0 (t ≈ 0.76850 and 1/t ≈ 1.3012). To prove this result, we made a full use of computer programs as follows: For the base algorithm that is designed in a conventional manner, we first construct an equivalent finite state diagram with about 300 states. Then for each state, we generate a finite set of inequalities such that the competitive ratio at that state is at most 1/t if the set of inequalities do not have a real solution. The latter can be checked by Mathematica. The number of inequalities generated was approximately 600 in total, and our computation time was 30 minutes using Athlon XP 2600+.

Year: 2009
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