ReLU Neural Networks of Polynomial Size for Exact Maximum Flow Computation

This paper studies the expressive power of artificial neural networks (NNs) with rectified linear units. To study them as a model of real-valued computation, we introduce the concept of Max-Affine Arithmetic Programs and show equivalence between them and NNs concerning natural complexity measures. We then use this result to show that two fundamental combinatorial optimization problems can be solved with polynomial-size NNs, which is equivalent to the existence of very special strongly polynomial time algorithms. First, we show that for any undirected graph with $n$ nodes, there is an NN of size $\mathcal{O}(n^3)$ that takes the edge weights as input and computes the value of a minimum spanning tree of the graph. Second, we show that for any directed graph with $n$ nodes and $m$ arcs, there is an NN of size $\mathcal{O}(m^2n^2)$ that takes the arc capacities as input and computes a maximum flow. These results imply in particular that the solutions of the corresponding parametric optimization problems where all edge weights or arc capacities are free parameters can be encoded in polynomial space and evaluated in polynomial time, and that such an encoding is provided by an NN.