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 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.

Quantum and classical algorithms for approximate submodular function minimization

Submodular functions are set functions mapping every subset of some ground set of size n into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization~\cite{LSW15} runs in time \so{n^3 \cdot \eo + n^4} where \eo denotes the cost to evaluate the function on any set. For functions with range [-1,1], the best \eps-additive approximation algorithm~\cite{CLSW17} runs in time \so{n^{5/3}/\eps^{2} \cdot \eo}. In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of \cite{CLSW17} and runs in time \so{n^{3/2}/\eps^2 \cdot \eo}. Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time \so{n^{5/4}/\eps^{5/2} \cdot \log(1/\eps) \cdot \eo}. The main ingredient of the quantum result is a new method for sampling with high probability T independent elements from any discrete probability distribution of support size n in time \bo{\sqrt{Tn}}. Previous quantum algorithms for this problem were of complexity \bo{T\sqrt{n}}.