# Lexicographically optimal earliest arrival flows

### 抄録

A dynamic network introduced by Ford and Fulkerson is a directed graph in which each arc has a capacity and a transit time. The evacuation problem is one of the fundamental problems in a dynamic network. The goal of this problem is to find the minimum time limit Θ such that we can send all the supplies to the sinks within time Θ. An earliest arrival flow is an optimal flow for the evacuation problem such that the amount of supplies which have reached the sinks is maximized at every time step. It is known that in a dynamic network with multiple sinks, if the sinks have capacities, then an earliest arrival flow does not necessarily exist. In this paper, to cope with this issue, we first introduce a lexicographically optimal earliest arrival flow in a dynamic network with multiple sinks. Then we propose a pseudo-polynomial-time algorithm for finding a lexicographically optimal earliest arrival flow. Furthermore, we prove that if the transit time of every arc is zero, then we can find a lexicographically optimal earliest arrival flow in polynomial time.

元の言語 英語 18-33 16 Networks 75 1 https://doi.org/10.1002/net.21902 出版済み - 1 1 2020

Polynomials
Directed graphs

### All Science Journal Classification (ASJC) codes

• Software
• Information Systems
• Hardware and Architecture
• Computer Networks and Communications

### これを引用

：: Networks, 巻 75, 番号 1, 01.01.2020, p. 18-33.

title = "Lexicographically optimal earliest arrival flows",
abstract = "A dynamic network introduced by Ford and Fulkerson is a directed graph in which each arc has a capacity and a transit time. The evacuation problem is one of the fundamental problems in a dynamic network. The goal of this problem is to find the minimum time limit Θ such that we can send all the supplies to the sinks within time Θ. An earliest arrival flow is an optimal flow for the evacuation problem such that the amount of supplies which have reached the sinks is maximized at every time step. It is known that in a dynamic network with multiple sinks, if the sinks have capacities, then an earliest arrival flow does not necessarily exist. In this paper, to cope with this issue, we first introduce a lexicographically optimal earliest arrival flow in a dynamic network with multiple sinks. Then we propose a pseudo-polynomial-time algorithm for finding a lexicographically optimal earliest arrival flow. Furthermore, we prove that if the transit time of every arc is zero, then we can find a lexicographically optimal earliest arrival flow in polynomial time.",
author = "Naoyuki Kamiyama",
year = "2020",
month = "1",
day = "1",
doi = "10.1002/net.21902",
language = "English",
volume = "75",
pages = "18--33",
journal = "Networks",
issn = "0028-3045",
publisher = "Wiley-Liss Inc.",
number = "1",

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TY - JOUR

T1 - Lexicographically optimal earliest arrival flows

AU - Kamiyama, Naoyuki

PY - 2020/1/1

Y1 - 2020/1/1

N2 - A dynamic network introduced by Ford and Fulkerson is a directed graph in which each arc has a capacity and a transit time. The evacuation problem is one of the fundamental problems in a dynamic network. The goal of this problem is to find the minimum time limit Θ such that we can send all the supplies to the sinks within time Θ. An earliest arrival flow is an optimal flow for the evacuation problem such that the amount of supplies which have reached the sinks is maximized at every time step. It is known that in a dynamic network with multiple sinks, if the sinks have capacities, then an earliest arrival flow does not necessarily exist. In this paper, to cope with this issue, we first introduce a lexicographically optimal earliest arrival flow in a dynamic network with multiple sinks. Then we propose a pseudo-polynomial-time algorithm for finding a lexicographically optimal earliest arrival flow. Furthermore, we prove that if the transit time of every arc is zero, then we can find a lexicographically optimal earliest arrival flow in polynomial time.

AB - A dynamic network introduced by Ford and Fulkerson is a directed graph in which each arc has a capacity and a transit time. The evacuation problem is one of the fundamental problems in a dynamic network. The goal of this problem is to find the minimum time limit Θ such that we can send all the supplies to the sinks within time Θ. An earliest arrival flow is an optimal flow for the evacuation problem such that the amount of supplies which have reached the sinks is maximized at every time step. It is known that in a dynamic network with multiple sinks, if the sinks have capacities, then an earliest arrival flow does not necessarily exist. In this paper, to cope with this issue, we first introduce a lexicographically optimal earliest arrival flow in a dynamic network with multiple sinks. Then we propose a pseudo-polynomial-time algorithm for finding a lexicographically optimal earliest arrival flow. Furthermore, we prove that if the transit time of every arc is zero, then we can find a lexicographically optimal earliest arrival flow in polynomial time.

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U2 - 10.1002/net.21902

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