Smooth Routing in Decaying Trees

TL;DR

This paper investigates the complex problem of smoothly scheduling evacuation paths in trees under time constraints, proving NP-hardness and providing ILP solutions for optimal evacuation timing.

Contribution

It establishes NP-hardness for the evacuation scheduling problem in trees and introduces an ILP approach to determine the latest evacuation time.

Findings

NP-hardness proven for trees, stars, and paths

ILP effectively computes optimal evacuation times

Relaxation of ILP offers insights into problem complexity

Abstract

Motivated by evacuation scenarios arising in extreme events such as flooding or forest fires, we study the problem of smoothly scheduling a set of paths in graphs where connections become impassable at some point in time. A schedule is smooth if no two paths meet on an edge and the number of paths simultaneously located at a vertex does not exceed its given capacity. We study the computational complexity of the problem when the underlying graph is a tree, in particular a star or a path. We prove that already in these settings, the problem is NP-hard even with further restrictions on the capacities or on the time when all connections ceased. We provide an integer linear program (ILP) to compute the latest possible time to evacuate. Using the ILP and its relaxation, we solve sets of artificial (where each underlying graph forms either a path or star) and semi-artificial instances (where the graphs are obtained from German cities along rivers), study the runtimes, and compare the results of the ILP with those of its relaxation.