Sink Location Problems in Dynamic Flow Grid Networks

TL;DR

This paper addresses the problem of optimally locating evacuation sinks in dynamic flow grid networks to minimize evacuation time, proving polynomial-time solutions for grid networks with uniform capacities and transit times.

Contribution

It introduces a polynomial-time algorithm for the 1-sink location problem specifically in grid networks with uniform edge capacities and transit times.

Findings

Polynomial-time solution for 1-sink location in grid networks.

Extension of sink location problem to more complex network classes.

Improved understanding of evacuation optimization in grid networks.

Abstract

A dynamic flow network consists of a directed graph, where nodes called sources represent locations of evacuees, and nodes called sinks represent locations of evacuation facilities. Each source and each sink are given supply representing the number of evacuees and demand representing the maximum number of acceptable evacuees, respectively. Each edge is given capacity and transit time. Here, the capacity of an edge bounds the rate at which evacuees can enter the edge per unit time, and the transit time represents the time which evacuees take to travel across the edge. The evacuation completion time is the minimum time at which each evacuees can arrive at one of the evacuation facilities. Given a dynamic flow network without sinks, once sinks are located on some nodes or edges, the evacuation completion time for this sink location is determined. We then consider the problem of locating sinks to minimize the evacuation completion time, called the sink location problem. The problems have been given polynomial-time algorithms only for limited networks such as paths, cycles, and trees, but no polynomial-time algorithms are known for more complex network classes. In this paper, we prove that the 1-sink location problem can be solved in polynomial-time when an input network is a grid with uniform edge capacity and transit time.