The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition problem for Generic Search is NP-complete. We utilize this finding to strengthen a complexity result from order theory. Given a partial order π and a set of triples, the NP-complete intermezzo problem asks for a linear extension of π where each first element of a triple is not between the other two. We show that this problem remains NP-complete even when the Hasse diagram of the partial order forms a tree of bounded height. In contrast, we give an XP-algorithm for the problem when parameterized by the width of the partial order. Furthermore, we show that - under the assumption of the Exponential Time Hypothesis - the running time of this algorithm is asymptotically optimal. LIPIcs, Vol. 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), pages 22:1-22:18

Abstract Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.

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 in 1985, where the author showed that this problem is efficiently solvable for depth first search (DFS) trees and breadth first search (BFS) trees. If one defines such a search tree in the same way as done for BFS, i.e., by connecting every vertex to its first neighbor, then we call this an ${\cal F}$-tree. If, on the other hand, we connect it with its most recently visited neighbor (as in DFS) we call this an ${\cal L}$-tree. In this paper, we consider related search paradigms. We prove that the search tree problem can be solved in polynomial time for ${\cal L}$-trees of lexicographic depth first search, whereas the ${\cal F}$-tree recognition problem is $\mathcal{NP}$-complete for lexicographic breadth first search, lexicographic depth first search, maximum cardinality search, and maximal neighborhood search. Furthermore, we present polynomial results for both types of trees on chordal graphs.

Lexicographic Depth First Search (LexDFS) is a special variant of a Depth First Search (DFS), which was introduced by Corneil and Krueger in 2008. While this search has been used in various applications, in contrast to other graph searches, no general linear time implementation is known to date. In 2014, Köhler and Mouatadid achieved linear running time to compute some special LexDFS orders for cocomparability graphs. In this paper, we present a linear time implementation of LexDFS for chordal graphs. Our algorithm is able to find any LexDFS order for this graph class. To the best of our knowledge this is the first unrestricted linear time implementation of LexDFS on a non-trivial graph class. In the algorithm we use a search tree computed by Lexicographic Breadth First Search (LexBFS).

In order to promote public transport many municipalities use traffic signal control with a priority for buses or trams. In this paper, we address the problem of finding optimal passive transit signal priority settings. Building on a cyclically time-expanded network model for the combined traffic assignment traffic signal coordination problem, we introduce a suitable queuing model and several modifications to model public transport vehicles appropriately. We evaluate the applicability of this approach by computing and analyzing optimal solutions for several instances of a real-world scenario.

In several applications of network flows, additional constraints have to be considered. In this paper, we study flows, where the flow particles have an orientation. For example, cargo containers with doors only on one side and train coaches with 1st and 2nd class compartments have such an orientation. If the end position has a mandatory orientation, not every path from source to sink is feasible for routing or additional transposition maneuvers have to be made. As a result, a source-sink path may visit a certain vertex several times. We describe structural properties of optimal solutions, determine the computational complexity, and present an approach for approximating such flows.

Most approaches for optimizing traffic signal timings deal with the daily traffic. However, there are a few occasional events like football matches or concerts of musicians that lead to exceptional traffic situations. Still, such events occur more or less regularly and place and time are known in advance. Hence, it is possible to anticipate such events with special signal timings. In this paper, we present an extension of a cyclically time-expanded network flow model and a corresponding mixed-integer linear programming formulation for simultaneously optimizing traffic signal timings and traffic assignment for such events. Besides the mathematical analysis of this approach, we demonstrate its capabilities by computing signal timings for a real world scenario.

We consider a constrained shortest path problem with the possibility to refill the resource at certain nodes. This problem is motivated by routing electric vehicles with a comparatively short cruising range due to the limited battery capacity. Thus, for longer distances the battery has to be recharged on the way. Furthermore, electric vehicles can recuperate energy during downhill drive. We extend the common constrained shortest path problem to arbitrary costs on edges and we allow regaining resources at the cost of higher travel time. We show that this yields not shortest paths but shortest walks that may contain an arbitrary number of cycles. We study the structure of optimal solutions and develop approximation algorithms for finding short walks under mild assumptions on charging functions. We also address a corresponding network flow problem that generalizes these walks.

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 article, 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. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 244–261 2015

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.