# Flow-weighted Layered Metric Euclidean Capacitated Steiner Tree Problem

## TL;DR

This paper introduces the FLaMECaST problem, a layered Euclidean Steiner tree variant with capacity constraints, proves its NP-hardness, and offers a polynomial-time approximation algorithm under specific conditions.

## Contribution

It defines the novel FLaMECaST problem, proves its NP-hardness, and develops a dynamic programming approximation method for special cases.

## Key findings

- FLaMECaST is NP-hard to approximate.
- A dynamic programming approach achieves a near-optimal approximation in polynomial time for certain instances.
- The method extends to sources on convex polygons.

## Abstract

Motivated by hierarchical networks, we introduce the Flow-weighted Layered Metric Euclidean Capacitated Steiner Tree (FLaMECaST) problem, a variant of the Euclidean Steiner tree with layered structure and capacity constraints per layer. The goal is to construct a cost-optimal Steiner forest connecting a set of sources to a set of sinks under load-dependent edge costs. We prove that FLaMECaST is NP-hard to approximate, even in restricted cases where all sources lie on a circle. However, assuming few additional constraints for such instances, we design a dynamic program that achieves a $\left(1 + \frac{1}{2^n}\right)$-approximation in polynomial time. By generalizing the structural insights the dynamic program is based on, we extend the approach to certain settings, where all sources are positioned on a convex polygon.