Online linear optimization over permutations

Authors:
Shota Yasutake;Kohei Hatano;Shuji Kijima;Eiji Takimoto;Masayuki Takeda
Affiliations:
Department of Informatics, Kyushu University, Japan;Department of Informatics, Kyushu University, Japan;Department of Informatics, Kyushu University, Japan;Department of Informatics, Kyushu University, Japan;Department of Informatics, Kyushu University, Japan
Venue:
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Year:
2011

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Abstract

This paper proposes an algorithm for online linear optimization problem over permutations; the objective of the online algorithm is to find a permutation of {1,…,n} at each trial so as to minimize the "regret" for T trials. The regret of our algorithm is $O(n^2 \sqrt{T \ln n})$ in expectation for any input sequence. A naive implementation requires more than exponential time. On the other hand, our algorithm uses only O(n) space and runs in O(n2) time in each trial. To achieve this complexity, we devise two efficient algorithms as subroutines: One is for minimization of an entropy function over the permutahedronPn, and the other is for randomized rounding over Pn.