We consider a dynamic model of traffic that has received a lot of attention in the past few years. Infinitesimally small agents aim to travel from a source to a destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form based on the deterministic queueing model whenever the inflow into a link exceeds its capacity.Are equilibria in this model meaningful as a prediction of traffic behavior? For this to be the case, a certain notion of stability under ongoing perturbations is needed. Real traffic consists of discrete, atomic “packets”, rather than being a continuous flow of non-atomic agents. Users may not choose an absolutely quickest route available, if there are multiple routes with very similar travel times. We would hope that in both these situations - a discrete packet model, with packet size going to 0, and $\varepsilon$-equilibria, with $\varepsilon$ - going to 0 - equilibria converge to dynamic equilibria in the flow over time model. No such convergence results were known.We show that such a convergence result does hold in single-commodity instances for both of these settings, in a unified way. More precisely, we introduce a notion of “strict” $\varepsilon$-equilibria, and show that these must converge to the exact dynamic equilibrium in the limit as $\varepsilon \rightarrow 0$. We then show that results for the two settings mentioned can be deduced from this with only moderate further technical effort.

We consider two key issues faced by many clustering methods when used for data summarization, namely (a) an unfair representation of "demographic groups” and (b) distorted summarizations, where data points in the summary represent subsets of the original data of vastly different sizes. Previous work made important steps towards handling separately each of these two issues in the context of the fundamental k-Center clustering objective through the study of fast algorithms for natural models that address them. We show that it is possible to effectively address both (a) and (b) simultaneously by presenting a clustering procedure that works for a canonical combined model and (i) is fast, both in theory and practice, (ii) exhibits a worst-case constant-factor guarantee, and (iii) gives promising computational results showing that there can be significant benefits in addressing both issues together instead of sequentially.

We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form whenever the inflow into a link exceeds its capacity. Despite lots of interest, some very basic questions remain open in this model. We resolve a number of them: • We show uniqueness of journey times in equilibria. • We show continuity of equilibria: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions. • We demonstrate that, assuming constant inflow into the network at the source, equilibria always settle down into a “steady state” in which the behavior extends forever in a linear fashion. One of our main conceptual contributions is to show that the answer to the first two questions, on uniqueness and continuity, are intimately connected to the third. Our result also shows very clearly that resolving uniqueness and continuity, despite initial appearances, cannot be resolved by analytic techniques, but are related to very combinatorial aspects of the model. To resolve the third question, we substantially extend the approach of Cominetti et al. [1], who show a steady-state result in the regime where the input flow rate is smaller than the network capacity. The full version of this extended abstract can be found on the arXiv preprint server as article 2111.06877

The mathematical approaches for modeling dynamic traffic can roughly be divided into two categories: discrete packet routing models and continuous flow over time models. Despite very vital research activities on models in both categories, the connection between these approaches was poorly understood so far. In this work we build this connection by specifying a (competitive) packet routing model, which is discrete in terms of flow and time, and by proving its convergence to the intensively studied model of flows over time with deterministic queuing. More precisely, we prove that the limit of the convergence process, when decreasing the packet size and time step length in the packet routing model, constitutes a flow over time with multiple commodities. In addition, we show that the convergence result implies the existence of approximate equilibria in the competitive version of the packet routing model. This is of significant interest as exact pure Nash equilibria, similar to almost all other competitive models, cannot be guaranteed in the multi-commodity setting. Moreover, the introduced packet routing model with deterministic queuing is very application-oriented as it is based on the network loading module of the agent-based transport simulation MATSim. As the present work is the first mathematical formalization of this simulation, it provides a theoretical foundation and an environment for provable mathematical statements for MATSim.

The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly $r$ times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of $r$-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex $n$-gon, the flip graph of plane trees on an arbitrary set of $n$ points, and the flip graph of noncrossing perfect matchings on a set of $n$ points in convex position. In addition, we consider two flip graphs on classes of nongeometric objects: the flip graph of permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of $\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and nonexistence of rainbow cycles for different values of $r$, $n$, and $k$.

Modeling traffic in road networks is a widely studied but challenging problem, especially under the assumption that drivers act selfishly. A common approach used in simulation software is the deterministic queuing model, for which the structure of dynamic equilibria has been studied extensively in the last couple of years. The basic idea is to model traffic by a continuous flow that travels over time from a source to a sink through a network, in which the arcs are endowed with transit times and capacities. Whenever the flow rate exceeds the capacity a queue builds up and the infinitesimally small flow particles wait in line in front of the bottleneck. Since the queues have no physical dimension, it was not possible, until now, to represent spillback in this model. This was a big drawback, since spillback can be regularly observed in real traffic situations and has a huge impact on travel times in highly congested regions. We extend the deterministic queuing model by introducing a storage capacity that bounds the total amount of flow on each arc. If an arc gets full, the inflow capacity is reduced to the current outflow rate, which can cause queues on previous arcs and blockages of intersections, i.e., spillback. We carry over the main results of the original model to our generalization and characterize dynamic equilibria, called Nash flows over time, by sequences of particular static flows, we call spillback thin flows. Furthermore, we give a constructive proof for the existence of dynamic equilibria, which suggests an algorithm for their computation. This solves an open problem stated by Koch and Skutella in 2010 [13].

Nash flows over time describe the behavior of selfish users eager to reach their destination as early as possible while traveling along the arcs of a network with capacities and transit times. Throughout the past decade, they have been thoroughly studied in single-source single-sink networks for the deterministic queuing model, which is of particular relevance and frequently used in the context of traffic and transport networks. In this setting there exist Nash flows over time that can be described by a sequence of static flows featuring special properties, so-called `thin flows with resetting'. This insight can also be used algorithmically to compute Nash flows over time. We present an extension of these result to networks with multiple sources and sinks which are much more relevant in practical applications. In particular, we come up with a subtle generalization of thin flows with resetting which yields a compact description as well as an algorithmic approach for computing multi-terminal Nash flows over time.

Mobile users of maps typically need detailed information about their surroundings plus some context information about remote places. In order to avoid that the map partly gets too dense, cartographers have designed mapping functions that enlarge a user-defined focus region--such functions are sometimes called fish-eye projections. The extra map space occupied by the enlarged focus region is compensated by distorting other parts of the map. We argue that, in a map showing a network of roads relevant to the user, distortion should preferably take place in those areas where the network is sparse. Therefore, we do not apply a predefined mapping function. Instead, we consider the road network as a graph whose edges are the road segments. We compute a new spatial mapping with a graph-based optimization approach, minimizing the square sum of distortions at edges. Our optimization method is based on a convex quadratic program (CQP); CQPs can be solved in polynomial time. Important requirements on the output map are expressed as linear inequalities. In particular, we show how to forbid edge crossings. We have implemented our method in a prototype tool. For instances of different sizes, our method generated output maps that were far less distorted than those generated with a predefined fish-eye projection. Future work is needed to automate the selection of roads relevant to the user. Furthermore, we aim at fast heuristics for application in real-time systems.