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

Yu Chen, Zihan Tan

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.

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

Abstract

We study the following distance realization problem. Given a quasi-metric on a set of terminals, does there exist a directed Okamura-Seymour graph that realizes as the (directed) shortest-path distance metric on ? 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 satisfy Monge property, if one exists. Combined with our result above, this gives an efficient algorithm for the distance realization problem.

Paper Structure

This paper contains 39 sections, 3 theorems, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work.
  4. Technical Overview
  5. Preliminaries
  6. Demands and routings.
  7. Partial quasi-metrics.
  8. Preparation: Nests, Edge-weight LP, and Restricting Pairs
  9. Graph structure: nests and Path Insertion
  10. Nests.
  11. Path Insertion.
  12. Setting the edge weights: LP and its dual
  13. Restricting Pairs.
  14. Completing the Proof of \ref{['thm: directed 4 point condition']}
  15. Good sequences and good pairs
  16. ...and 24 more sections

Key Result

Theorem 1

A quasi-metric $D$ on $T$ is realizable by a directed Okamura-Seymour instance where the terminals in $T$ lie on the boundary in the circular ordering $\sigma$, iff for all quadruples $t_1,t_2,t_3,t_4$ of terminals in $T$ appearing in $\sigma$ in this order, $D(t_1,t_3)+D(t_2,t_4)\ge D(t_1,t_4)+D(t_

Figures (7)

  • Figure 1: An illustration of an incorrect way of inserting path $\pi_{a,b}$ under certain $(D,\sigma)$.
  • Figure 2: An illustration of intersecting forbidden areas (yellow is partially covered by blue).
  • Figure 3: An illustration of a path insertion process and its produced nest.
  • Figure 4: An illustration of complete and incomplete walls.
  • Figure 5: An illustration of switching in Case 2.2 and Case 3.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Claim 3
  • Claim 4
  • proof
  • Claim 6
  • Lemma 7
  • Claim 8
  • proof
  • Claim 9
  • ...and 4 more