103 research outputs found
Computing earliest arrival flows with multiple sources
Earliest arrival flows are motivated by applications related to
evacuation. Given a network with capacities and transit times on
the arcs, a subset of source nodes with supplies and a sink node,
the task is to send the given supplies from the sources to the sink
"as quickly as possible". The latter requirement is made more
precise by the earliest arrival property which requires that the
total amount of flow that has arrived at the sink is maximal for all
points in time simultaneously.
It is a classical result from the 1970s that, for the special case
of a single source node, earliest arrival flows do exist and can be
computed by essentially applying the Successive Shortest Path
Algorithm for min-cost flow computations. While it has previously
been observed that an earliest arrival flow still exists for
multiple sources, the problem of computing one efficiently has been
open. We present an exact algorithm for this problem whose running
time is strongly polynomial in the input plus output size of the
problem
Optimal rounding of instantaneous fractional flows over time
"August 1999."Includes bibliographical references (p. 10-11).by Lisa K. Fleischer [and] James B. Orlin
Transshipment Problem and Its Variants: A Review
The transshipment problem is a unique Linear Programming Problem (LLP) in that it considers the assumption that all sources and sinks can both receive and distribute shipments at the same time (function in both directions). Being an extension of the classical transportation problem, the transshipment problem covers a wide range of scenarios for logistics and/or transportation inputs and products and offers optimum alternatives for same. In this work the review of literatures from the origin and current trends on the transshipment problem were carried out. This was done in view of the unique managerial needs and formulation of models/objective functions. It was revealed that the LLP offers a wide range of decision alternative for the operations manager based on the dynamic and challenging nature of logistics management. Key words: Transshipment problem, Linear Programming Problems (LPP), model, objective functions, decision alternative
Quickest Flows Over Time
Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in timeâexpanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the timeâexpanded network. We present several approaches for coping with this difficulty. First, inspired by the work of Ford and Fulkerson on maximal sâtâflows over time (or âmaximal dynamic sâtâflowsâ), we show that static lengthâbounded flows lead to provably good multicommodity flows over time. Second, we investigate âcondensedâ timeâexpanded networks which rely on a rougher discretization of time. We prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed timeâexpanded network of polynomial size. In particular, our approach yields fully polynomialâtime approximation schemes for the NPâhard quickest minâcost and multicommodity flow problems. For single commodity problems, we show that storage of flow at intermediate nodes is unnecessary, and our approximation schemes do not use any
Sink Location Problems in Dynamic Flow Grid Networks
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.Comment: 16 pages, 6 figures, full version of a paper accepted at COCOON 202
Multi-agent Path Planning and Network Flow
This paper connects multi-agent path planning on graphs (roadmaps) to network
flow problems, showing that the former can be reduced to the latter, therefore
enabling the application of combinatorial network flow algorithms, as well as
general linear program techniques, to multi-agent path planning problems on
graphs. Exploiting this connection, we show that when the goals are permutation
invariant, the problem always has a feasible solution path set with a longest
finish time of no more than steps, in which is the number of
agents and is the number of vertices of the underlying graph. We then give
a complete algorithm that finds such a solution in time, with
being the number of edges of the graph. Taking a further step, we study time
and distance optimality of the feasible solutions, show that they have a
pairwise Pareto optimal structure, and again provide efficient algorithms for
optimizing two of these practical objectives.Comment: Corrected an inaccuracy on time optimal solution for average arrival
tim
Routing Diverse Evacuees with Cognitive Packets
This paper explores the idea of smart building evacuation when evacuees can
belong to different categories with respect to their ability to move and their
health conditions. This leads to new algorithms that use the Cognitive Packet
Network concept to tailor different quality of service needs to different
evacuees. These ideas are implemented in a simulated environment and evaluated
with regard to their effectiveness.Comment: 7 pages, 7 figure
Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems
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, hop-routing 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 NP-hard.
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 single-sink 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 Non-Approximability: 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
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