Böther, Maximilian; Schiller, Leon; Fischbeck, Philipp; Molitor, Louise; Krejca, Martin S.; Friedrich, Tobias Evolutionary Minimization of Traffic CongestionIEEE Transactions on Evolutionary Computation 2023: 1809–1821
Traffic congestion is a major issue that can be solved by suggesting drivers alternative routes they are willing to take. This concept has been formalized as a strategic routing problem in which a single alternative route is suggested to an existing one. We extend this formalization and introduce the Multiple-Routes problem, which is given a start and destination and aims at finding up to \(n\) different routes that the drivers strategically disperse over, minimizing the overall travel time of the system. Due to the NP-hard nature of the problem, we introduce the Multiple-Routes evolutionary algorithm (MREA) as a heuristic solver. We study several mutation and crossover operators and evaluate them on real-world data of Berlin, Germany. We find that a combination of all operators yields the best result, reducing the overall travel time by a factor between \(1.8\) and \(3\), in the median, compared to all drivers taking the fastest route. For the base case \(n = 2\), we compare our MREA to the highly tailored optimal solver by Bläsius et al. (ATMOS 2020), and show that, in the median, our approach finds solutions of quality at least \(99.69 \%\) of an optimal solution while only requiring \(40 \%\) of the time.
Bläsius, Thomas; Fischbeck, Philipp; Friedrich, Tobias; Katzmann, Maximilian Solving Vertex Cover in Polynomial Time on Hyperbolic Random GraphsTheory of Computing Systems 2023: 28–51
The computational complexity of the VertexCover problem has been studied extensively. Most notably, it is NP-complete to find an optimal solution and typically NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice.
Khomutovskiy, Ivan; Dunker, Rebekka; Dierking, Jessica; Egbert, Julian; Helms, Christian; Schöllkopf, Finn; Casel, Katrin; Fischbeck, Philipp; Friedrich, Tobias; Isaac, Davis; Krogmann, Simon; Lenzner, Pascal Applying Skeletons to Speed Up the Arc-Flags Routing AlgorithmSIAM Symposium on Algorithm Engineering and Experiments (ALENEX) 2023: 110–122
The Single-Source Shortest Path problem is classically solved by applying Dijkstra's algorithm. However, the plain version of this algorithm is far too slow for real-world applications such as routing in large road networks. To amend this, many speed-up techniques have been developed that build on the idea of computing auxiliary data in a preprocessing phase, that is used to speed up the queries. One well-known example is the Arc-Flags algorithm that is based on the idea of precomputing edge flags to make the search more goal-directed. To explain the strong practical performance of such speed-up techniques, several graph parameters have been introduced. The skeleton dimension is one such parameter that has already been used to derive runtime bounds for some speed-up techniques. Moreover, it was experimentally shown to be low in real-world road networks. We introduce a method to incorporate skeletons, the underlying structure behind the skeleton dimension, to improve routing speed-up techniques even further. As a proof of concept, we develop new algorithms called SKARF and SKARF+ that combine skeletons with Arc-Flags, and demonstrate via extensive experiments on large real-world road networks that SKARF+ yields a significant reduction of the search space and the query time of about 30% to 40% over Arc-Flags. We also prove theoretical bounds on the query time of SKARF, which is the first time an Arc-Flags variant has been analyzed in terms of skeleton dimension.
Cohen, Sarel; Fischbeck, Philipp; Friedrich, Tobias; Krejca, Martin S. The Common-Neighbors Metric is Noise-Robust and Reveals Substructures of Real-World NetworksPacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD) 2023: 67–79
Real-world networks typically display a complex structure that is hard to explain by a single model. A common approach is to partition the edges of the network into disjoint simpler structures. An important property in this context is locality—incident vertices usually have many common neighbors. This allows to classify edges into two groups, based on the number of the common neighbors of their incident vertices. Formally, this is captured by the common-neighbors (CN) metric, which forms the basis of many metrics for detecting outlier edges. Such outliers can be interpreted as noise or as a substructure. We aim to understand how useful the metric is, and empirically analyze several scenarios. We randomly insert outlier edges into real-world and generated graphs with high locality, and measure the metric accuracy for partitioning the combined edges. In addition, we use the metric to decompose real-world networks, and measure properties of the partitions. Our results show that the CN metric is a very good classifier that can reliably detect noise up to extreme levels (\(83\%\) noisy edges). We also provide mathematically rigorous analyses on special random-graph models. Last, we find the CN metric consistently decomposes real-world networks into two graphs with very different structures.