arXiv subjects

Svenja M. Griesbach

Publications and source records attributed to Svenja M. Griesbach.

9 recordsLinked to original sources

arXiv2026

Carbon Pricing in Traffic Networks

We study how externality pricing can steer traffic toward Wardrop equilibria respecting prescribed budgets. Our multi-commodity model encompasses carbon pricing, cordon pricing, and tradable credit schemes. We characterize the flows attainable when one common price may be assigned to each externality. We show that for continuous travel times and flow-independent externalities, every feasible externality budget is implementable. For flow-dependent externalities, smoothness bounds quantify the necessary budget relaxation; this links budget implementability to the price of anarchy. If travel times and externalities admit convex potentials, we give conditions for constructing prices implementing a given budget through convex optimization and specify explicit prices ensuring that relaxed budgets are respected by every equilibrium. As, for a single separable constant externality, total externality is nonincreasing in its price, this enables us to compute the whole price--equilibrium curve in finite time for separable, continuous, strictly increasing, piecewise affine travel times. Although this curve may have exponentially many pieces, the minimum price implementing a feasible budget is computable in polynomial time. The same machinery computes the range of market-clearing prices for tradable credit schemes. A computational study on several realistic traffic networks illustrates emission reductions from carbon pricing and the effects of cordon pricing on emissions within the cordon.

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arXiv2026

Approximation Algorithms for Discounted Graph Search with Norm Objectives

We introduce a unified framework for classical search and routing problems, including pathwise search, expanding search, the minimum spanning tree problem, and the traveling salesperson problem. The framework is based on two parameters. The first is a discount factor $α\in [0,1]$: the first traversal of an edge incurs its full cost, whereas each subsequent traversal incurs only an $α$-fraction of this cost. For a path starting at a designated root vertex, the $α$-latency of a vertex is the discounted cost accumulated until the vertex is first visited. The second parameter is a norm parameter $p\geq 1$. The objective is to find a root-starting path that visits all vertices and minimizes the $p$-norm of the resulting vector of $α$-latencies. The model interpolates between several well-studied objectives. For $p=1$ and $α=1$, it recovers pathwise search; for $p=1$ and $α=0$, it recovers expanding search. As $p$ tends to infinity, the objective converges to a makespan-type criterion. At the endpoints $α=1$ and $α=0$, this limiting objective corresponds to TSP-type and MST-type behavior, respectively. For $p=1$, we give polynomial-time constant-factor approximation algorithms for all $α\in[0,1]$, matching the best known guarantees for expanding search at $α=0$ and pathwise search at $α=1$. For general $p\geq 1$, we obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.

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arXiv2025

Public Signals in Network Congestion Games

We consider a largely untapped potential for the improvement of traffic networks that is rooted in the inherent uncertainty of travel times. Travel times are subject to stochastic uncertainty resulting from various parameters such as weather condition, occurrences of road works, or traffic accidents. Large mobility services have an informational advantage over single network users as they are able to learn traffic conditions from data. A benevolent mobility service may use this informational advantage in order to steer the traffic equilibrium into a favorable direction. The resulting optimization problem is a task commonly referred to as signaling or Bayesian persuasion. Previous work has shown that the underlying signaling problem can be NP-hard to approximate within any non-trivial bounds, even for affine cost functions with stochastic offsets. In contrast, we show that in this case, the signaling problem is easy for many networks. We tightly characterize the class of single-commodity networks, in which full information revelation is always an optimal signaling strategy. Moreover, we construct a reduction from optimal signaling to computing an optimal collection of support vectors for the Wardrop equilibrium. For two states, this insight can be used to compute an optimal signaling scheme. The algorithm runs in polynomial time whenever the number of different supports resulting from any signal distribution is bounded to a polynomial in the input size. Using a cell decomposition technique, we extend the approach to a polynomial-time algorithm for multi-commodity parallel link networks with a constant number of commodities, even when we have a constant number of different states of nature.

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arXiv2025

Online Proportional Apportionment

Traditionally, the problem of apportioning the seats of a legislative body has been viewed as a one-shot process with no dynamic considerations. While this approach is reasonable for some settings, dynamic aspects play an important role in many others. We initiate the study of apportionment problems in an online setting. Specifically, we introduce a framework for proportional apportionment with no information about the future. In this model, time is discrete and there are $n$ parties that receive a certain share of the votes at each time step. An online algorithm needs to irrevocably assign a prescribed number of seats at each time, ensuring that each party receives its fractional share rounded up or down, and that the cumulative number of seats allocated to each party remains close to its cumulative share up to that time. We study deterministic and randomized online apportionment methods. For deterministic methods, we construct a family of adversarial instances that yield a lower bound, linear in $n$, on the worst-case deviation between the seats allocated to a party and its cumulative share. We show that this bound is best possible and is matched by a natural greedy method. As a consequence, a method guaranteeing that the cumulative number of seats assigned to each party up to any step equals its cumulative share rounded up or down (global quota) exists if and only if $n\leq 3$. Then, we turn to randomized allocations and show that, for $n\leq 3$, we can randomize over methods satisfying global quota with the additional guarantee that each party receives, in expectation, its proportional share in every step. Our proof is constructive: Any method satisfying these properties can be obtained from a flow on a recursively constructed network. We showcase the applicability of our results to obtain approximate solutions in the context of online dependent rounding procedures.

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arXiv2025

Improved Approximation Algorithms for the Expanding Search Problem

A searcher is tasked with exploring a graph with edge lengths and vertex weights, starting from a designated vertex. Initially, only the starting vertex is considered explored. At each step, the searcher adds an edge to the solution, connecting an unexplored vertex to an explored one. The time required to add an edge equals its length. The objective is to minimize the weighted sum of exploration times for all vertices. We demonstrate that this problem is hard to approximate and present algorithms with improved approximation guarantees. Specifically, we provide a $(2\mathrm{e} + \varepsilon)$-approximation for any $\varepsilon > 0$ for the general case. On instances where the vertex weights are binary, we achieve a $2\mathrm{e}$-approximation. Finally, we develop a polynomial-time approximation scheme (PTAS) for Euclidean graphs. Previously, only an $8$-approximation was known for all these cases.

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arXiv2024

Deterministic Impartial Selection with Weights

In the impartial selection problem, a subset of agents up to a fixed size $k$ among a group of $n$ is to be chosen based on votes cast by the agents themselves. A selection mechanism is impartial if no agent can influence its own chance of being selected by changing its vote. It is $α$-optimal if, for every instance, the ratio between the votes received by the selected subset is at least a fraction of $α$ of the votes received by the subset of size $k$ with the highest number of votes. We study deterministic impartial mechanisms in a more general setting with arbitrarily weighted votes and provide the first approximation guarantee, roughly $1/\lceil 2n/k\rceil$. When the number of agents to select is large enough compared to the total number of agents, this yields an improvement on the previously best known approximation ratio of $1/k$ for the unweighted setting. We further show that our mechanism can be adapted to the impartial assignment problem, in which multiple sets of up to $k$ agents are to be selected, with a loss in the approximation ratio of $1/2$.

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arXiv2024

Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem

We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial $(α,μ)$-approximation is possible, i.e., a solution that with budget $B+α$ for all $B \in \mathbb{R}_{\geq 0}$ is a multiplicative $μ$-approximation compared to the optimum solution with budget $B$. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a $(χ,1)$-approximation, where $χ$ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is $(γ,2)$-competitive where $γ$ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a $(γ,3)$-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a $(3χ,8)$-approximation and, more generally, a $\smash{\bigl((4\ell - 1)χ, \frac{2^{\ell + 2}}{2^{\ell}-1}\bigr)}$-approximation for every fixed $\ell \in \mathbb{N}$.

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arXiv2023

Information Design for Congestion Games with Unknown Demand

We study a novel approach to information design in the standard traffic model of network congestion games. It captures the natural condition that the demand is unknown to the users of the network. A principal (e.g., a mobility service) commits to a signaling strategy, observes the realized demand and sends a (public) signal to agents (i.e., users of the network). Based on the induced belief about the demand, the users then form an equilibrium. We consider the algorithmic goal of the principal: Compute a signaling scheme that minimizes the expected total cost of the induced equilibrium. We concentrate on single-commodity networks and affine cost functions, for which we obtain the following results. First, we devise a fully polynomial-time approximation scheme (FPTAS) for the case that the demand can only take two values. It relies on several structural properties of the cost of the induced equilibrium as a function of the updated belief about the distribution of demands. We show that this function is piecewise linear for any number of demands, and monotonic for two demands. Second, we give a complete characterization of the graph structures for which it is optimal to fully reveal the information about the realized demand. This signaling scheme turns out to be optimal for all cost functions and probability distributions over demands if and only if the graph is series-parallel. Third, we propose an algorithm that computes the optimal signaling scheme for any number of demands whose time complexity is polynomial in the number of supports that occur in a Wardrop equilibrium for some demand. Finally, we conduct a computational study that tests this algorithm on real-world instances.

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arXiv2023

Optimizing Throughput and Makespan of Queuing Systems by Information Design

We study the optimal provision of information for two natural performance measures of queuing systems: throughput and makespan. A set of parallel links is equipped with deterministic capacities and stochastic travel times where the latter depend on a realized scenario. A continuum of flow particles arrives at the system at a constant rate. A system operator knows the realization of the scenario and may (partially) reveal this information via a public signaling scheme to the flow particles. Upon arrival, the flow particles observe the signal issued by the system operator, form an updated belief about the realized scenario, and decide on a link to use. Inflow into a link exceeding the link's capacity builds up in a queue that increases the travel time on the link. Dynamic inflow rates are in a Bayesian dynamic equilibrium when the expected travel time along all links with positive inflow is equal at every point in time. We provide an additive polynomial time approximation scheme (PTAS) that approximates the optimal throughput by an arbitrary additive constant $ε>0$. The algorithm solves a Langrangian dual of the signaling problem with the Ellipsoid method whose separation oracle is implemented by a cell decomposition technique. We also provide a multiplicative fully polynomial time approximation scheme (FPTAS) that does not rely on strong duality and, thus, allows to compute also the optimal signals. It uses a different cell decomposition technique together with a piece-wise convex under-estimator of the optimal value function. Finally, we consider the makespan of a Bayesian dynamic equilibrium which is defined as the last point in time when a total given value of flow leaves the system. Using a variational inequality argument, we show that full information revelation is a public signaling scheme that minimizes the makespan.

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