This paper studies the expressive power of artificial neural networks with rectified linear units. In order 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 neural networks concerning natural complexity measures. We then use this result to show that two fundamental combinatorial optimization problems can be solved with polynomial-size neural networks. First, we show that for any undirected graph with $n$ nodes, there is a neural network (with fixed weights and biases) 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 a neural network of size $\mathcal{O}(m^2n^2)$ that takes the arc capacities as input and computes a maximum flow. Our results imply that these two problems can be solved with strongly polynomial time algorithms that solely use affine transformations and maxima computations, but no comparison-based branchings.