Thomas Lücking

Computing Nash equilibria for scheduling on restricted parallel links

We consider the problem of routing n users on m parallel links, under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. Thus, the problem is equivalent to the correspondingly restricted problem of assigning n jobs to m parallel machines. In a pure Nash equilibrium, no user may improve its own individual cost (delay) by unilaterally switching to another link from its set of allowed links. As our main result, we introduce a polynomial time algorithm to compute from any given assignment a pure Nash equilibrium with non-increased makespan. The algorithm gradually changes a given assignment by pushing unsplittable user traffics through a network that is defined by the users and the links. Here, we use ideas from blocking flows. Furthermore, we use similar techniques as in the generic Preflow-Push algorithm to approximate a schedule with minimum makespan, gaining an improved approximation factor of 2 - 1/w 1 for identical links, where w 1 is the largest user traffic. We extend this result to related links, gaining an approximation factor of 2. Our approximation algorithms run in polynomial time. We close with tight upper bounds on the coordination ratio for pure Nash equilibria.

Nashification and the coordination ratio for a selfish routing game

We study the problem of n users selfishly routing traffic through a network consisting of m parallel related links. Users route their traffic by choosing private probability distributions over the links with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy. Nashification is the problem of converting any given non-equilibrium routing into a Nash equilibrium without increasing the social cost. Our first result is an O(nm2) time algorithm for Nashification. This algorithm can be used in combination with any approximation algorithm for the routing problem to compute a Nash equilibrium of the same quality. In particular, this approach yields a PTAS for the computation of a best Nash equilibrium. Furthermore, we prove a lower bound of Ω(2√n) and an upper bound of O(2n) for the number of greedy selfish steps for identical link capacities in the worst case. In the second part of the paper we introduce a new structural parameter which allows us to slightly improve the upper bound on the coordination ratio for pure Nash equilibria in [3]. The new bound holds for the individual coordination ratio and is asymptotically tight. Additionally, we prove that the known upper bound of 1+√4m-3/2 on the coordination ratio for pure Nash equilibria also holds for the individual coordination ratio in case of mixed Nash equilibria, and we determine the range of m for which this bound is tight.

A New Model for Selfish Routing

In this work, we introduce and study a new model for selfish routing over non-cooperative networks that combines features from the two such best studied models, namely the KP model and the Wardrop model in an interesting way.

Selfish Routing in Non-cooperative Networks: A Survey

We study the problem of n users selfishly routing traffics through a shared network. Users route their traffics by choosing a path from their source to their destination of the traffic with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy.

Which Is the Worst-Case Nash Equilibrium?

A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium – the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9].

The price of anarchy for polynomial social cost

In this work, we consider an interesting variant of the well studied KP model for selfish routing on parallel links, which reflects some influence from the much older Wardrop model [J.G. Wardrop, Some theoretical aspects of road traffic research, Proc. Inst. of Civil Eng. Part II 1 (1956) 325-378]. In the new model, user traffics are still unsplittable and links are identical. Social cost is now the expectation of the sum, over all links, of latency costs; each latency cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link. The resulting social cost is called polynomial social cost, or monomial social cost when the latency cost function is a monomial. All considered polynomials are of degree d, where d ≥ 2, and have non-negative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the monomial price of anarchy (MPoA) and the polynomial price of anarchy (PPoA) as our evaluation measures. Through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the Stirling numbers of the second kind and the Bell numbers, we obtain a multitude of results: • For the special case of identical users: The fully mixed Nash equilibrium, where all probabilities are strictly positive, maximizes polynomial social cost. The MPoA is no more than Bd, the Bell number of order d. This immediately implies that the PPoA is no more than Σ1 ≤ t ≤ dBt. For the special case of two links, the MPoA is no more than 2d-2(1 + (1/n)d-1), and this bound is tight for n = 2. • The MPoA is exactly ((2d - 1)d/(d - 1)(2d - 2)d-1)((d - 1)/d)d for pure Nash equilibria. This immediately implies that the PPoA is no more than Σ2 ≤ t ≤ d ((2t - 1)t/(t - 1)(2t - 2)t-1)((t - 1)/t)t.

Structure and complexity of extreme Nash equilibria

We study extreme Nash equilibria in the context of a selfish routing game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1 + e) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.Considering pure Nash equilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is N P-hard to approximate the worst social cost within a multiplicative factor better than 2 - 2/(m + 1).

Subresultants revisited

Subresultants and polynomial remainder sequences are an important tool in polynomial computer algebra. In this survey, we sketch the history, formalize a unified framework for the various notions, derive a number of results from the early 1970s within our framework, and report on implementations.

The price of anarchy for restricted parallel links

In the model of restricted parallel links, n users must be routed on m parallel links under the restriction that the link for each user be chosen from a certain set of allowed links for the user. In a (pure) Nash equilibrium, no user may improve its own Individual Cost (latency) by unilaterally switching to another link from its set of allowed links. The Price of Anarchy is a widely adopted measure of the worst-case loss (relative to optimum) in system performance (maximum latency) incurred in a Nash equilibrium. In this work, we present a comprehensive collection of bounds on Price of Anarchy for the model of restricted parallel links and for the special case of pure Nash equilibria. Specifically, we prove: • For the case of identical users and identical links, the Price of Anarchy is

Extreme Nash Equilibria

\Omega (\frac{{\rm lg}\,m}{{\rm lg\,lg}\,m})