Geometric rescaling algorithms for submodular function minimization

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