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

Maximum Contraflow Evacuation Planning Problems On Multi-network

Contraflow approach for the evacuation planning problem increases outbound capacity of the evacuation routes by the reversal of anti-parallel arcs, if such arcs exist. The existing literature focuses on network contraflow problems that allow only anti-parallel arcs with equal transit time. However, the problems modeled on multi-network, allowing parallel as well as anti-parallel arcs with not necessarily equal transit time, seem more realistic. In this paper, we study the maximum dynamic contraflow problem for multi-network and propose efficient solution techniques to them with discrete as well as continuous time settings. We also extend the results to solve earliest version of the problem for two terminal series parallel (TTSP) multi-network

An Integer Network Flow Problem with Bridge Capacities

In this paper a modified version of dynamic network ows is discussed. Whereas dynamic network flows are widely analyzed already, we consider a dynamic flow problem with aggregate arc capacities called Bridge Problem which was introduced by Melkonian [Mel07]. We extend his research to integer flows and show that this problem is strongly NP-hard. For practical relevance we also introduce and analyze the hybrid bridge problem, i.e. with underlying networks whose arc capacity can limit aggregate flow (bridge problem) or the flow entering an arc at each time (general dynamic flow). For this kind of problem we present efficient procedures for special cases that run in polynomial time. Moreover, we present a heuristic for general hybrid graphs with restriction on the number of bridge arcs. Computational experiments show that the heuristic works well, both on random graphs and on graphs modeling also on realistic scenarios

Complexity of the Temporal Shortest Path Interdiction Problem

In the shortest path interdiction problem, an interdictor aims to remove arcs of total cost at most a given budget from a directed graph with given arc costs and traversal times such that the length of a shortest s-t-path is maximized. For static graphs, this problem is known to be strongly NP-hard, and it has received considerable attention in the literature. While the shortest path problem is one of the most fundamental and well-studied problems also for temporal graphs, the shortest path interdiction problem has not yet been formally studied on temporal graphs, where common definitions of a "shortest path" include: latest start path (path with maximum start time), earliest arrival path (path with minimum arrival time), shortest duration path (path with minimum traveling time including waiting times at nodes), and shortest traversal path (path with minimum traveling time not including waiting times at nodes). In this paper, we analyze the complexity of the shortest path interdiction problem on temporal graphs with respect to all four definitions of a shortest path mentioned above. Even though the shortest path interdiction problem on static graphs is known to be strongly NP-hard, we show that the latest start and the earliest arrival path interdiction problems on temporal graphs are polynomial-time solvable. For the shortest duration and shortest traversal path interdiction problems, however, we show strong NP-hardness, but we obtain polynomial-time algorithms for these problems on extension-parallel temporal graphs

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 Generalized Notion of Time for Modeling Temporal Networks

Most approaches for modeling and analyzing temporal networks do not explicitly discuss the underlying notion of time. In this paper, we therefore introduce a generalized notion of time for temporal networks. Our approach also allows for considering non-deterministic time and incomplete data, two issues that are often found when analyzing data-sets extracted from online social networks, for example. In order to demonstrate the consequences of our generalized notion of time, we also discuss the implications for the computation of (shortest) temporal paths in temporal networks

The Dynamics of Internet Traffic: Self-Similarity, Self-Organization, and Complex Phenomena

The Internet is the most complex system ever created in human history. Therefore, its dynamics and traffic unsurprisingly take on a rich variety of complex dynamics, self-organization, and other phenomena that have been researched for years. This paper is a review of the complex dynamics of Internet traffic. Departing from normal treatises, we will take a view from both the network engineering and physics perspectives showing the strengths and weaknesses as well as insights of both. In addition, many less covered phenomena such as traffic oscillations, large-scale effects of worm traffic, and comparisons of the Internet and biological models will be covered.Comment: 63 pages, 7 figures, 7 tables, submitted to Advances in Complex System