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Traditionally, the coordination of multiple traffic signals and the traffic assignment problem in an urban street network are considered as two separate optimization problems. However, it is easy to see that the traffic assignment has an influence on the optimal signal coordination and, vice versa, a change in the signal coordination changes the optimal traffic assignment. In this paper we present a cyclically time-expanded network and a corresponding mixed integer linear programming formulation for simultaneously optimizing both the coordination of traffic signals and the traffic assignment in an urban street network. Although the new cyclically time-expanded network provides a model of both traffic and signals close to reality, it still has the advantage of a linear objective function. Using this model we compute optimized signal coordinations and traffic assignment on real-world street networks. To evaluate the practical relevance of the computed solutions we conduct extensive simulation experiments using two established traffic simulation tools that reveal the advantages of our model.

In this paper, we present a cyclically time-expanded network model for simultaneous optimization of traffic assignment and traffic signal parameters, in particular offsets, split times, and phase orders. Since travel times are of great importance for developing realistic solutions for traffic assignment and traffic signal coordination in urban road networks, we perform an extensive analysis of the model. We show that a linear time-expanded model can reproduce realistic travel times especially for use with traffic signals and we verify this by simulation. Furthermore, we show how exact mathematical programming techniques can be used for optimizing the control of traffic signals. We provide computational results for real world instances and demonstrate the capabilities of the cyclically time-expanded by simulation results obtained with state-of-the-art traffic simulation tools. MotivationTraffic signals can be seen as a backbone in the control of traffic flows in urban areas. Since the appearance of the first signalized intersections over 100 years ago, the improvement of signal control strategies has been an important subject for research. Friedrich [13] and Papageorgiou et al. [24] provide overviews of the main research lines over the last years.Recently, we presented a model for traffic signal coordination based on cyclically timeexpanded networks (see [20,21,32]). In contrast to most of the previous mathematical approaches, we focused not only on optimizing the offsets of signals, but also optimizing the implied traffic assignment. Each change in the signal parameters may influence the travel times in the network. Consequently, road users will quickly adapt to a new signal coordination and they will switch to faster routes if they suffer from the last intervention 1 arXiv:1509.08709v1 [cs.DM] 29 Sep 2015 in the signal plans. The traffic assignment, i.e., the distribution of traffic in the road network, may change significantly.Contrary to previous practical approaches based on non-linear models and heuristics, we focus on a model which allows for the use of exact programming techniques. Hence, proving optimality of a solution or bounding the gap between primal and dual solution was given a higher priority than modeling every effect of real traffic. Thus, this paper is particularly addressing the underlying theoretical questions of traffic signal optimization and the corresponding hardness.In this paper, we present an extensive analysis of the cyclically time-expanded model. It is a hybrid between a static, i.e., time independent, and a dynamic, i.e., time dependent, network flow approach. Although this time-expanded model is based on a linear program, it can reproduce very realistic non-linear flow-dependent travel times for urban traffic networks with signal control. In particular, both typical convex link performance functions and the time-dependent behavior of traffic, necessary for traffic signal coordination, are realistically mapped in this model. Additionally, the propagation of platoons of cars that is ...

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Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show N P-completeness. We complement our results by providing linear time algorithms for these searches on split graphs. * The work of this paper was done in the framework of a bilateral project between Brandenburg