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 $\frac{1}{3}$ of the flow and this bound is tight. For the special case of networks with a single source or sink, this fraction is $\frac12$ 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

Selfish Routing on Dynamic Flows

Selfish routing on dynamic flows over time is used to model scenarios that vary with time in which individual agents act in their best interest. In this paper we provide a survey of a particular dynamic model, the deterministic queuing model, and discuss how the model can be adjusted and applied to different real-life scenarios. We then examine how these adjustments affect the computability, optimality, and existence of selfish routings.Comment: Oberlin College Computer Science Honors Thesis. Supervisor: Alexa Sharp, Oberlin Colleg

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

Multi-Source Multi-Sink Nash Flows over Time

Nash flows over time describe the behavior of selfish users eager to reach their destination as early as possible while traveling along the arcs of a network with capacities and transit times. Throughout the past decade, they have been thoroughly studied in single-source single-sink networks for the deterministic queuing model, which is of particular relevance and frequently used in the context of traffic and transport networks. In this setting there exist Nash flows over time that can be described by a sequence of static flows featuring special properties, so-called `thin flows with resetting\u27. This insight can also be used algorithmically to compute Nash flows over time. We present an extension of these results to networks with multiple sources and sinks which are much more relevant in practical applications. In particular, we come up with a subtle generalization of thin flows with resetting, which yields a compact description as well as an algorithmic approach for computing multi-terminal Nash 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 time‐expanded 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 time‐expanded network. We present several approaches for coping with this difficulty. First, inspired by the work of Ford and Fulkerson on maximal s‐t‐flows over time (or “maximal dynamic s‐t‐flows”), we show that static length‐bounded flows lead to provably good multicommodity flows over time. Second, we investigate “condensed” time‐expanded 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 time‐expanded network of polynomial size. In particular, our approach yields fully polynomial‐time approximation schemes for the NP‐hard quickest min‐cost 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

A Stackelberg Strategy for Routing Flow over Time

Routing games are used to to understand the impact of individual users' decisions on network efficiency. Most prior work on routing games uses a simplified model of network flow where all flow exists simultaneously, and users care about either their maximum delay or their total delay. Both of these measures are surrogates for measuring how long it takes to get all of a user's traffic through the network. We attempt a more direct study of how competition affects network efficiency by examining routing games in a flow over time model. We give an efficiently computable Stackelberg strategy for this model and show that the competitive equilibrium under this strategy is no worse than a small constant times the optimal, for two natural measures of optimality