48 research outputs found
Computing all Wardrop Equilibria parametrized by the Flow Demand
We develop an algorithm that computes for a given undirected or directed
network with flow-dependent piece-wise linear edge cost functions all Wardrop
equilibria as a function of the flow demand. Our algorithm is based on
Katzenelson's homotopy method for electrical networks. The algorithm uses a
bijection between vertex potentials and flow excess vectors that is piecewise
linear in the potential space and where each linear segment can be interpreted
as an augmenting flow in a residual network. The algorithm iteratively
increases the excess of one or more vertex pairs until the bijection reaches a
point of non-differentiability. Then, the next linear region is chosen in a
Simplex-like pivot step and the algorithm proceeds. We first show that this
algorithm correctly computes all Wardrop equilibria in undirected
single-commodity networks along the chosen path of excess vectors. We then
adapt our algorithm to also work for discontinuous cost functions which allows
to model directed edges and/or edge capacities. Our algorithm is
output-polynomial in non-degenerate instances where the solution curve never
hits a point where the cost function of more than one edge becomes
non-differentiable. For degenerate instances we still obtain an
output-polynomial algorithm computing the linear segments of the bijection by a
convex program. The latter technique also allows to handle multiple
commodities
The price of anarchy in routing games as a function of the demand
The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite
The Network Improvement Problem for Equilibrium Routing
In routing games, agents pick their routes through a network to minimize
their own delay. A primary concern for the network designer in routing games is
the average agent delay at equilibrium. A number of methods to control this
average delay have received substantial attention, including network tolls,
Stackelberg routing, and edge removal.
A related approach with arguably greater practical relevance is that of
making investments in improvements to the edges of the network, so that, for a
given investment budget, the average delay at equilibrium in the improved
network is minimized. This problem has received considerable attention in the
literature on transportation research and a number of different algorithms have
been studied. To our knowledge, none of this work gives guarantees on the
output quality of any polynomial-time algorithm. We study a model for this
problem introduced in transportation research literature, and present both
hardness results and algorithms that obtain nearly optimal performance
guarantees.
- We first show that a simple algorithm obtains good approximation guarantees
for the problem. Despite its simplicity, we show that for affine delays the
approximation ratio of 4/3 obtained by the algorithm cannot be improved.
- To obtain better results, we then consider restricted topologies. For
graphs consisting of parallel paths with affine delay functions we give an
optimal algorithm. However, for graphs that consist of a series of parallel
links, we show the problem is weakly NP-hard.
- Finally, we consider the problem in series-parallel graphs, and give an
FPTAS for this case.
Our work thus formalizes the intuition held by transportation researchers
that the network improvement problem is hard, and presents topology-dependent
algorithms that have provably tight approximation guarantees.Comment: 27 pages (including abstract), 3 figure
Network improvement for equilibrium routing
Routing games are frequently used to model the behavior of traffic in large networks, such as road networks. In transportation research, the problem of adding capacity to a road network in a cost-effective manner to minimize the total delay at equilibrium is known as the Network Design Problem, and has received considerable attention. However, prior to our work, little was known about guarantees for polynomial-time algorithms for this problem. We obtain tight approximation guarantees for general and series-parallel networks, and present a number of open questions for future work
Side-Constrained Dynamic Traffic Equilibria
We study dynamic traffic assignment with side-constraints. We first give a
counter-example to a key result from the literature regarding the existence of
dynamic equilibria for volume-constrained traffic models in the classical
edge-delay model. Our counter-example shows that the feasible flow space need
not be convex and it further reveals that classical infinite dimensional
variational inequalities are not suited for the definition of side-constrained
dynamic equilibria. We propose a new framework for side-constrained dynamic
equilibria based on the concept of feasible -deviations of flow
particles in space and time. Under natural assumptions, we characterize the
resulting equilibria by means of quasi-variational and variational
inequalities, respectively. Finally, we establish first existence results for
side-constrained dynamic equilibria for the non-convex setting of
volume-constraints.Comment: 57 pages, 8 figure