71 research outputs found
Graph Orientation and Flows Over Time
Flows over time are used to model many real-world logistic and routing
problems. The networks underlying such problems -- streets, tracks, etc. -- are
inherently undirected and directions are only imposed on them to reduce the
danger of colliding vehicles and similar problems. Thus the question arises,
what influence the orientation of the network has on the network flow over time
problem that is being solved on the oriented network. In the literature, this
is also referred to as the contraflow or lane reversal problem.
We introduce and analyze the price of orientation: How much flow is lost in
any orientation of the network if the time horizon remains fixed? We prove that
there is always an orientation where we can still send of the
flow and this bound is tight. For the special case of networks with a single
source or sink, this fraction is which is again tight. We present
more results of similar flavor and also show non-approximability results for
finding the best orientation for single and multicommodity maximum flows over
time
Optimal rounding of instantaneous fractional flows over time
"August 1999."Includes bibliographical references (p. 10-11).by Lisa K. Fleischer [and] James B. Orlin
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
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
An annotated overview of dynamic network flows
The need for more realistic network models led to the development of the dynamic network flow theory. In dynamic flow models it takes time for the flow to pass an arc, the flow can be delayed at nodes, and the network parameters, e.g., the arc capacities, can change in time. Surprisingly perhaps, despite being closer to reality, dynamic flow models have been overshadowed by the classical, static model. This is largely due to the fact that while very efficient solution methods exist for static flow problems, dynamic flow problems have proved to be more difficult to solve. Our purpose with this overview is to compensate for this eclipse and introduce dynamic flows to the interested reader. To this end, we present the main flow problems that can appear in a dynamic network, and review the literature for existing results about them. Our approach is solution oriented, as opposed to dealing with modelling issues. We intend to provide a survey that can be a first step for readers wondering whether a given dynamic network flow problem has been solved or not. Besides restating the problems, we also describe the main proposed solution methods. An additional feature of this paper is an annotated list of the most important references about the subject
Earliest Arrival Flows with Multiple Sources
Earliest arrival flows capture the essence of evacuation planning. 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 for many years. We present an exact algorithm for this problem whose running time is strongly polynomial in the input plus output size of the problem
The 1st International Electronic Conference on Algorithms
This book presents 22 of the accepted presentations at the 1st International Electronic Conference on Algorithms which was held completely online from September 27 to October 10, 2021. It contains 16 proceeding papers as well as 6 extended abstracts. The works presented in the book cover a wide range of fields dealing with the development of algorithms. Many of contributions are related to machine learning, in particular deep learning. Another main focus among the contributions is on problems dealing with graphs and networks, e.g., in connection with evacuation planning problems
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
Combinatorial Optimization
Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry
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