140 research outputs found
An FPTAS for Quickest Multicommodity Flows with Inflow-Dependent Transit Times
Given a network with capacities and transit times on the arcs, the quickest flow problem asks for a "flow over time" that satisfies given demands within minimal time. In the setting of flows over time, flow on arcs may vary over time and the transit time of an arc is the time it takes for flow to travel through this arc. In most real-world applications (such as, e.g., road traffic, communication networks, production systems, etc.), transit times are not fixed but depend on the current flow situation in the network. We consider the model where the transit time of an arc is given as a non-decreasing function of the rate of inflow into the arc. We prove that the quickest s-t-flow problem is NP-hard in this setting and give various approximation results, including a fully polynomial time approximation scheme (FPTAS) for the quickest multicommodity flow problem with bounded cos
Optimizing Emergency Transportation through Multicommodity Quickest Paths
In transportation networks with limited capacities and travel times on the arcs, a class of problems attracting a growing scientific interest is represented by the optimal routing and scheduling of given amounts of flow to be transshipped from the origin points to the specific destinations in minimum time. Such problems are of particular concern to emergency transportation where evacuation plans seek to minimize the time evacuees need to clear the affected area and reach the safe zones. Flows over time approaches are among the most suitable mathematical tools to provide a modelling representation of these problems from a macroscopic point of view. Among them, the Quickest Path Problem (QPP), requires an origin-destination flow to be routed on a single path while taking into account inflow limits on the arcs and minimizing the makespan, namely, the time instant when the last unit of flow reaches its destination. In the context of emergency transport, the QPP represents a relevant modelling tool, since its solutions are based on unsplittable dynamic flows that can support the development of evacuation plans which are very easy to be correctly implemented, assigning one single evacuation path to a whole population. This way it is possible to prevent interferences, turbulence, and congestions that may affect the transportation process, worsening the overall clearing time. Nevertheless, the current state-of-the-art presents a lack of studies on multicommodity generalizations of the QPP, where network flows refer to various populations, possibly with different origins and destinations. In this paper we provide a contribution to fill this gap, by considering the Multicommodity Quickest Path Problem (MCQPP), where multiple commodities, each with its own origin, destination and demand, must be routed on a capacitated network with travel times on the arcs, while minimizing the overall makespan and allowing the flow associated to each commodity to be routed on a single path. For this optimization problem, we provide the first mathematical formulation in the scientific literature, based on mixed integer programming and encompassing specific features aimed at empowering the suitability of the arising solutions in real emergency transportation plans. A computational experience performed on a set of benchmark instances is then presented to provide a proof-of-concept for our original model and to evaluate the quality and suitability of the provided solutions together with the required computational effort. Most of the instances are solved at the optimum by a commercial MIP solver, fed with a lower bound deriving from the optimal makespan of a splittable-flow relaxation of the MCQPP
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
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
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
An adaptive routing approach for personal rapid transit
Personal Rapid Transit (PRT) is a public transportation mode, in which small automated vehicles transport passengers on demand. Central control of the vehicles leads to interesting possibilities for optimized routings. The complexity of the involved routing problems together with the fact that routing algorithms for PRT essentially have to run in real-time often leads to the choice of fast greedy approaches. The most common routing approach is arguably a sequential one, where upcoming requests are greedily served in a quickest way without interfering with previously routed vehicles. The simplicity of this approach stems from the fact that a chosen route is never changed later. This is as well the main drawback of it, potentially leading to large detours. It is natural to ask how much one could gain by using a more adaptive routing strategy. This question is the main motivation of this article. In this paper, we first suggest a simple mathematical model for PRT, and then introduce a new adaptive routing algorithm that repeatedly uses solutions to an LP as a guide to route vehicles. Our routing approach incorporates new requests in the LP as soon as they appear, and reoptimizes the routing of all currently used vehicles, contrary to sequential routing. We provide preliminary computational results that give first evidence of the potential gains of an adaptive routing strategy, as used in our algorithm.National Science Foundation (U.S.) (Grants CCF-1115849 and CCF-0829878)United States. Office of Naval Research (Grants N00014-11-1-0053 and N00014-09-1-0326
Optimal rounding of instantaneous fractional flows over time
"August 1999."Includes bibliographical references (p. 10-11).by Lisa K. Fleischer [and] James B. Orlin
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
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