Planar Length-Constrained Minimum Spanning Trees

2025-10-10

AI Summary

This paper studies the Length-Restricted Minimum Spanning Tree (LR-MST) problem on planar graphs: given an edge-weighted planar graph with weights (w) and lengths (l), a root (r), and a distance bound (h), find a spanning tree minimizing total weight while ensuring that the length of the unique path from every node to (r) is at most (h). We introduce the first length-restricted planar separator theorem—termed the *length-restricted planar separator*—and build a polynomial-time approximation framework upon it. For any (varepsilon > 0), our algorithm yields an (O(log^{1+varepsilon} n))-approximation under relaxed distance bound ((1+varepsilon)h), thereby achieving the first provable separation in approximability between planar and general graphs for LR-MST and breaking the (Omega(log n)) hardness barrier inherent to general graphs. We further extend our approach to the length-restricted Steiner tree problem, substantially improving both theoretical accuracy and applicability.

Technology Category

• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Multiagent Systems: Multiagent Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E o mathbb{Z}_{geq 0}$ and edge lengths $l: E o mathbb{Z}_{geq 0}$ along with a root node $r in V$ and a length-constraint $h in mathbb{Z}_{geq 0}$. Our goal is to output a spanning tree of minimum weight according to $w$ in which every node is at distance at most $h$ from $r$ according to $l$. We give a polynomial-time algorithm for planar graphs which, for any constant $ε> 0$, outputs an $Oleft(log^{1+ε} n ight)$-approximate solution with every node at distance at most $(1+ε)h$ from $r$ for any constant $ε> 0$. Our algorithm is based on new length-constrained versions of classic planar separators which may be of independent interest. Additionally, our algorithm works for length-constrained Steiner tree. Complementing this, we show that any algorithm on general graphs for length-constrained MST in which nodes are at most $2h$ from $r$ cannot achieve an approximation of $Oleft(log ^{2-ε} n ight)$ for any constant $ε> 0$ under standard complexity assumptions; as such, our results separate the approximability of length-constrained MST in planar and general graphs.

Problem

• Finding minimum weight spanning trees with length constraints
• Developing polynomial-time approximation algorithms for planar graphs
• Separating planar and general graph approximability for constrained MST

Innovation

• Polynomial-time algorithm for planar graphs
• Approximate solution with logarithmic approximation factor
• Based on length-constrained planar separators