ReLU Neural Networks of Polynomial Size for Exact Maximum Flow Computation

TL;DR

This paper demonstrates that neural networks with ReLU units of polynomial size can exactly solve fundamental combinatorial optimization problems like maximum flow and minimum spanning tree, using only affine transformations and maxima.

Contribution

It introduces Max-Affine Arithmetic Programs and proves their equivalence to neural networks, enabling polynomial-size neural networks to solve key optimization problems exactly.

Findings

Neural networks of size O(n^3) can compute minimum spanning trees.

Neural networks of size O(m^2 n^2) can compute maximum flow.

These solutions do not rely on comparison-based branchings.

Abstract

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.