Maximum Weight Disjoint Paths in Outerplanar Graphs via Single-Tree Cut Approximators

TL;DR

This paper presents a constant-factor approximation algorithm for the maximum edge-disjoint paths problem in outerplanar graphs, advancing the understanding of routing problems in specific graph classes.

Contribution

It introduces a novel single-tree cut approximator for outerplanar graphs, enabling improved approximation algorithms for the maximum disjoint paths problem.

Findings

Provides a constant-factor approximation for ED in outerplanar graphs.

Develops a single-tree cut approximator for outerplanar graphs.

Extends previous results on cut approximators and routing problems.

Abstract

Since 1997 there has been a steady stream of advances for the maximum disjoint paths problem. Achieving tractable results has usually required focusing on relaxations such as: (i) to allow some bounded edge congestion in solutions, (ii) to only consider the unit weight (cardinality) setting, (iii) to only require fractional routability of the selected demands (the all-or-nothing flow setting). For the general form (no congestion, general weights, integral routing) of edge-disjoint paths ({\sc edp}) even the case of unit capacity trees which are stars generalizes the maximum matching problem for which Edmonds provided an exact algorithm. For general capacitated trees, Garg, Vazirani, Yannakakis showed the problem is APX-Hard and Chekuri, Mydlarz, Shepherd provided a $4$-approximation. This is essentially the only setting where a constant approximation is known for the general form of \textsc{edp}. We extend their result by giving a constant-factor approximation algorithm for general-form \textsc{edp} in outerplanar graphs. A key component for the algorithm is to find a {\em single-tree} $O(1)$ cut approximator for outerplanar graphs. Previously $O(1)$ cut approximators were only known via distributions on trees and these were based implicitly on the results of Gupta, Newman, Rabinovich and Sinclair for distance tree embeddings combined with results of Anderson and Feige.