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 a strongly polynomial exact SFM algorithm. 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 a O((n5⋅EO+n6)log2n) 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, yielding a simplified variant of their O(n3log2n⋅EO+n4logO(1)n) algorithm.

ACM-SIAM Symposium on Discrete Algorithms
Networks and Optimization

Dadush, D.N, Végh, L.A, & Zambelli, G. (2018). Geometric Rescaling Algorithms for Submodular Function Minimization. In Annual ACM-SIAM Symposium on Discrete Algorithms.