We consider two approaches that model timetable information in public transportation systems as shortest-path problems in weighted graphs. In the time-expanded approach, every event at a station, e.g., the departure of a train, is modeled as a node in the graph, while in the timedependent approach the graph contains only one node per station. Both approaches have been recently considered for (a simplified version of) the earliest arrival problem, but little is known about their relative performance. Thus far, there are only theoretical arguments in favor of the time-dependent approach. In this paper, we provide the first extensive experimental comparison of the two approaches. Using several real-world data sets, we evaluate the performance of the basic models and of several new extensions towards realistic modeling. Furthermore, new insights on solving bicriteria optimization problems in both models are presented. The time-expanded approach turns out to be more robust for modeling more complex scenarios, whereas the time-dependent approach shows a clearly better performance.

We describe a new technique for proving the existence of small -nets for hypergraphs satisfying certain simple conditions. The technique is particularly useful for proving o( 1 log 1 ) upper bounds which is not possible using the standard VC dimension theory. We apply the technique to several geometric hypergraphs and obtain simple proofs for the existence of O( 1 ) size -nets for them. This includes the geometric hypergraph in which the vertex set is a set of points in the plane and the hyperedges are defined by a set of pseudo-disks. This result was not known previously. We also get a very short proof for the existence of O( 1 ) size -nets for halfspaces in R 3 .

We continue the study of the effects of selfish behavior in the network design problem. We provide new bounds for the price of stability for network design with fair cost allocation for undirected graphs. We consider the most general case, for which the best known upper bound is the Harmonic number H n , where n is the number of agents, and the best previously known lower bound is 12/7 ≈ 1.778.We present a nontrivial lower bound of 42/23 ≈ 1.8261. Furthermore, we show that for two players, the price of stability is exactly 4/3, while for three players it is at least 74/48 ≈ 1.542 and at most 1.65. These are the first improvements on the bound of H n for general networks. In particular, this demonstrates a separation between the price of stability on undirected graphs and that on directed graphs, where H n is tight. Previously, such a gap was only known for the cases where all players have a shared source, and for weighted players.Keywords Algorithmic Game Theory ⋅ Price of Stability ⋅ Network Design A preliminary version of this work appeared in [8].

No abstract