Heuristic Optimal Transport in Branching Networks
M. Andrecut
TL;DR
The paper addresses the mismatch between classical OT, which yields non-branching, straight-line mappings, and real-world networks that heavily rely on branching. It introduces a fast, two-stage BOT framework: first solve a regularized OT (or LP) to obtain base associations, then apply a local, greedy-tabu branching optimization to construct a branching network that minimizes transport cost with a mass-transport exponent $\alpha\in[0,1]$, interpolating between OT ($\alpha=1$) and Steiner-type branching ($\alpha\approx0$). The method yields a deterministic algorithm with $O(N^2)$ complexity and demonstrates substantial cost reductions in synthetic, cardiovascular-like, and large-scale Santa Claus networks, including a practical demonstration via a web app. The work provides a scalable approach to designing branching transport networks with potential applications in logistics, biology, and urban planning.
Abstract
Optimal transport aims to learn a mapping of sources to targets by minimizing the cost, which is typically defined as a function of distance. The solution to this problem consists of straight line segments optimally connecting sources to targets, and it does not exhibit branching. These optimal solutions are in stark contrast with both natural, and man-made transportation networks, where branching structures are prevalent. Here we discuss a fast heuristic branching method for optimal transport in networks. We also provide several numerical applications to synthetic examples, a simplified cardiovascular network, and the "Santa Claus" distribution network which includes 141,182 cities around the world, with known location and population.