Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Planar Network Diversion

Matthias Bentert, Pål Grønås Drange, Fedor V. Fomin, Steinar Simonnes

TL;DR

This paper addresses computing robust diversions in planar networks by formulating Network Diversion as finding a minimal $s$-$t$-cut that includes a specified edge $b$, within $G$ planar and weighted. It introduces a deterministic $O(n \log n)$ algorithm built on the dual-graph correspondence between cuts and cycles and leverages Derigs' shortest odd path method to enforce parity constraints, ensuring the resulting cut is simple and minimal. The authors provide a concrete Rust implementation and show substantial practical speedups over prior MILP approaches, solving graphs with millions of vertices in seconds. The work advances planar-network resilience analysis and offers a solid foundation for future extensions to nonplanar graphs and crossing-number parametrizations, with potential impact on transportation planning and security analyses.

Abstract

Network Diversion is a graph problem that has been extensively studied in both the network-analysis and operations-research communities as a measure of how robust a network is against adversarial disruption. This problem is especially well motivated in transportation networks, which are often assumed to be planar. Motivated by this and recent theoretical advances for Network Diversion on planar input graphs, we develop a fast O(n log n) time algorithm and present a practical implementation of this algorithm that is able to solve instances with millions of vertices in a matter of seconds.

Planar Network Diversion

TL;DR

This paper addresses computing robust diversions in planar networks by formulating Network Diversion as finding a minimal --cut that includes a specified edge , within planar and weighted. It introduces a deterministic algorithm built on the dual-graph correspondence between cuts and cycles and leverages Derigs' shortest odd path method to enforce parity constraints, ensuring the resulting cut is simple and minimal. The authors provide a concrete Rust implementation and show substantial practical speedups over prior MILP approaches, solving graphs with millions of vertices in seconds. The work advances planar-network resilience analysis and offers a solid foundation for future extensions to nonplanar graphs and crossing-number parametrizations, with potential impact on transportation planning and security analyses.

Abstract

Network Diversion is a graph problem that has been extensively studied in both the network-analysis and operations-research communities as a measure of how robust a network is against adversarial disruption. This problem is especially well motivated in transportation networks, which are often assumed to be planar. Motivated by this and recent theoretical advances for Network Diversion on planar input graphs, we develop a fast O(n log n) time algorithm and present a practical implementation of this algorithm that is able to solve instances with millions of vertices in a matter of seconds.

Paper Structure

This paper contains 12 sections, 7 theorems, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Results
  3. Background
  4. Preliminaries and Notation
  5. Derigs' Shortest Odd Path
  6. An Application of Shortest Odd Path
  7. Network Diversion on Planar Graphs
  8. Algorithm for Planar Network Diversion
  9. Implementation and Experimental Setup
  10. Experimental Results
  11. Conclusion
  12. Illustration of diverse cuts

Key Result

Proposition 1

A solution to Network Diversion on a planar graph $G$ corresponds directly to a minimum-weight, simple, odd-parity cycle $E^\star_C \supseteq \{ {b}^{*} \}$ in the dual graph of $G$, where parity is measured with respect to a simple $(s,t)$-reference path in $G$ that contains $b$.

Figures (5)

  • Figure 1: Shortest Detour Path reduced to Shortest Odd Path by subdividing all edges except the detour.
  • Figure 2: An illustration of the different steps in our algorithm for Network Diversion on planar graphs. We assume here that all edges have the same weight. Steps (a) -- (c) are directly from our algorithm and Step (d) shows the reduction from finding a shortest path that crosses the computed path an odd number of times to finding a shortest odd path given by \ref{['cor:path-odd-number-f']}. Step (e) shows the solution of running Derigs' algorithm and Step (f) shows the solution to the input instance we computed.
  • Figure 3: Running times of our implementation for Network Diversion on Delaunay graphs of different sizes. The x-axis shows the number of vertices in 1000s and the blue dots show the running time in milliseconds. The green line shows a running mean running time over the previous 15 instances. The orange line shows the line $n \log_2 n/100$ ms.
  • Figure 4: Comparison of the previous milp algorithm by Cullenbine et al. cullenbine2013theoreticalcomputational (blue, with filled circles for the values they provide and empty circles for interpolated values, and crosses for timeouts) and ours (orange) on grids of different sizes.
  • Figure 5: Diverse minimal $s$-$t$-cuts, with associated costs, computed on a Delaunay graph with 35 vertices and 91 edges.

Theorems & Definitions (9)

  • Proposition 1: Cullenbine et al. cullenbine2013theoreticalcomputational
  • Theorem 1: Derigs' algorithm derigs1985efficientdijkstralike
  • Theorem 2: diestel2016graphtheory
  • Corollary 1
  • Corollary 2
  • proof
  • Lemma 1: Folklore
  • Theorem 3
  • proof