The Shortcut Problem - Complexity and Algorithms

We study a graph-augmentation problem arising from a technique applied in recent approaches for route planning. Many such methods enhance the graph by inserting shortcuts, i.e., additional edges (u,v) such that the length of (u,v) is the distance from u to v. Given a weighted, directed graph G and a number c ∈ Z>0, the shortcut problem asks how to insert c shortcuts into G such that the expected number of edges that are contained in an edge-minimal shortest path from a random node s to a random node t is minimal. In this work, we study the algorithmic complexity of the problem and give approximation algorithms for a special graph class. Further, we state ILP-based exact approaches and show how to stochastically evaluate a given shortcut assignment on graphs that are too large to do so exactly. Submitted: March 2011 Reviewed: November 2011 Revised: April 2012 Accepted: June 2012 Final: July 2012 Published: August 2012 Article type: Regular paper Communicated by: S. Albers Partially supported by the DFG (project WA654/16-1). E-mail addresses: [email protected] (Reinhard Bauer) gianlorenzo.d [email protected] (Gianlorenzo D’Angelo) [email protected] (Daniel Delling) [email protected] (Andrea Schumm) dorothea.wagner@kit (Dorothea Wagner) 448 Bauer et al. The Shortcut Problem – Complexity and Algorithms