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Geometric rescaling algorithms for submodular function minimization

Dadush, Daniel, Végh, László A. and Zambelli, Giacomo (2018) Geometric rescaling algorithms for submodular function minimization. In: Czumaj, Artur, (ed.) Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. Proceedings. Society for industrial and applied mathematics, Philadelphia, USA, pp. 832-848. ISBN 978-611975031

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Identification Number: 10.1137/1.9781611975031.54

Abstract

We present a new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our new algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and the Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we use the geometric rescaling technique, which has recently gained attention in linear programming. We adapt this technique to SFM and obtain a weakly polynomial bound O((n4 · EO + n5) log(nL)). Secondly, we exhibit a general combinatorial black-box approach to turn any strongly polynomial εL-approximate SFM oracle into an strongly polynomial exact SFM algorithm. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudopolynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige-Wolfe algorithm. Combined with the geometric rescaling technique, the black-box approach provides a O((n5 · EO + n6) log2 n) algorithm. Finally, we show that one of the techniques we develop in the paper, “sliding”, can also be combined with the cutting-plane method of Lee, Sidford, and Wong [27], yielding a simplified variant of their O(n3 log2 n · EO + n4 logO(1) n) algorithm

Item Type: Book Section
Official URL: http://doi.org/10.1137/1.9781611975031.54
Additional Information: © 2018 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 02 Feb 2018 14:33
Last Modified: 20 Feb 2019 06:34
Projects: 639.071.510, EP/M02797X/1
Funders: NWO, EPSRC
URI: http://eprints.lse.ac.uk/id/eprint/86635

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