The VPN Tree Routing Conjecture for Outerplanar Networks

TL;DR

This paper proves that outerplanar networks satisfy the Pyramidal Routing Conjecture, which implies the VPN Tree Routing Conjecture, advancing understanding of optimal tree-like solutions in network design.

Contribution

The paper introduces new tools and proves the Pyramidal Routing Conjecture for outerplanar networks, supporting the VPN Tree Routing Conjecture.

Findings

Outerplanar networks satisfy the PR Conjecture.

New general tools for network analysis are developed.

Supports the VPN Tree Routing Conjecture in specific network classes.

Abstract

The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of the VPN Conjecture for rings was found a few months ago by Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In their paper, Grandoni et al. introduce another conjecture, called the Pyramidal Routing Conjecture (or simply PR Conjecture), which implies the VPN Conjecture. Here we consider a strengthened version of the PR Conjecture. First we establish several general tools which can be applied in arbitrary networks. Then we use them to prove that outerplanar networks satisfy the PR Conjecture.