Abstract
Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hoprouting of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NPhard.
One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing.
We investigate the singlesink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include:
a) Logarithmic NonApproximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an O(\log n) approximation factor, unless P=NP.
b) Polylogarithmic Bicriteria Approximations: Polynomial time (O(\log^8 n), O(\log^2 \kappa)) bicritera approximation algorithms for the Confluent Quickest Flow problem where \kappa is the number of sinks, in both directed and undirected graphs.
Corresponding results are also developed for the Confluent Maximum Flow over time problem. The techniques developed also improve recent approximation algorithms for static confluent flows.
BibTeX  Entry
@InProceedings{golin_et_al:LIPIcs:2017:8243,
author = {Mordecai J. Golin and Hadi Khodabande and Bo Qin},
title = {{Nonapproximability and Polylogarithmic Approximations of the SingleSink Unsplittable and Confluent Dynamic Flow Problems}},
booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)},
pages = {41:141:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770545},
ISSN = {18688969},
year = {2017},
volume = {92},
editor = {Yoshio Okamoto and Takeshi Tokuyama},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8243},
URN = {urn:nbn:de:0030drops82435},
doi = {10.4230/LIPIcs.ISAAC.2017.41},
annote = {Keywords: Optimization, Approximation, Dynamic Flow, Confluent Flow}
}
Keywords: 

Optimization, Approximation, Dynamic Flow, Confluent Flow 
Seminar: 

28th International Symposium on Algorithms and Computation (ISAAC 2017) 
Issue Date: 

2017 
Date of publication: 

04.12.2017 