Tropical Circuits with Scalar Multiplication Gates

2026-07-13

AI Summary

This work investigates the expressive power of tropical circuits—comprising max, addition, and positive scalar multiplication gates—in computing two fundamental combinatorial optimization problems: the maximum-weight directed spanning tree and the maximum-weight perfect matching in bipartite graphs. By leveraging tools from algebraic circuit complexity, tropical algebra, and combinatorial optimization, the study establishes, for the first time, exponential-size lower bounds for tropical circuits solving these problems. This result not only provides the first exponential lower bound on the size of tropical circuits but also reveals an exponential gap in expressive power between monotone and non-monotone maxout neural networks. Furthermore, it implies that input-convex neural networks may require exponentially larger architectures to represent the same functions.

Abstract

We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.

Problem

• tropical circuits
• scalar multiplication gates
• exponential size lower bounds
• input-convex neural networks
• maxout neural networks

Innovation

• tropical circuits
• scalar multiplication gates
• exponential lower bounds
• maxout neural networks
• input-convex neural networks