Paths and Intersections: Characterization of Quasi-metrics in Directed Okamura-Seymour Instances

TL;DR

This paper characterizes when a quasi-metric on a set of terminals can be realized by a directed Okamura-Seymour graph, introducing a Monge property condition and algorithms for construction and ordering.

Contribution

It generalizes previous undirected results to directed graphs, introduces a new graph analysis method based on paths and intersections, and provides algorithms for realization and ordering.

Findings

Monge property is necessary and sufficient for realization with boundary ordering.

A greedy algorithm constructs the realizing graph from the quasi-metric.

An efficient algorithm finds a boundary ordering satisfying the Monge property.

Abstract

We study the following distance realization problem. Given a quasi-metric $D$ on a set $T$ of terminals, does there exist a directed Okamura-Seymour graph that realizes $D$ as the (directed) shortest-path distance metric on $T$? We show that, if we are further given the circular ordering of terminals lying on the boundary, then Monge property is a sufficient and necessary condition. This generalizes previous results for undirected Okamura-Seymour instances. With the circular ordering, we give a greedy algorithm for constructing a directed Okamura-Seymour instance that realizes the input quasi-metric. The algorithm takes the dual perspective concerning flows and routings, and is based on a new way of analyzing graph structures, by viewing graphs as \emph{paths and their intersections}. We believe this new understanding is of independent interest and will prove useful in other problems in graph theory and graph algorithms. We also design an efficient algorithm for finding such a circular ordering that makes $D$ satisfy Monge property, if one exists. Combined with our result above, this gives an efficient algorithm for the distance realization problem.