Affiliation:
1. Centrum Wiskunde and Informatica, 1098 XG Amsterdam, Netherlands
2. London School of Economics and Political Science, London WC2A 2AE, United Kingdom
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 algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound [Formula: see text]. Second, we exhibit a general combinatorial black box approach to turn [Formula: see text]-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial 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 an [Formula: see text] algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their [Formula: see text] algorithm.
Publisher
Institute for Operations Research and the Management Sciences (INFORMS)
Subject
Management Science and Operations Research,Computer Science Applications,General Mathematics
Cited by
3 articles.
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