Geometric rescaling algorithms for submodular function minimization

Dadush, Daniel, Végh, László A. ORCID: 0000-0003-1152-200X 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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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
Date Deposited:	02 Feb 2018 14:33
Last Modified:	30 Mar 2024 08:09
Projects:	639.071.510, EP/M02797X/1
Funders:	NWO, EPSRC
URI:	http://eprints.lse.ac.uk/id/eprint/86635

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