In competitive packet routing games, packets are routed selfishly through a network and scheduling policies at edges determine which packages are forwarded first if there is not enough capacity on an edge to forward all packages at once. We analyze the impact of priority lists on the worst-case quality of pure Nash equilibria. A priority list is an ordered list of players that may or may not depend on the edge. Whenever the number of packets entering an edge exceeds the inflow capacity, packets are processed in list order. We derive several new bounds on the price of anarchy and stability for global and local priority policies. We also consider the question of the complexity of computing an optimal priority list. It turns out that even for very restricted cases, i.e., for routing on a tree, the computation of an optimal priority list is APX-hard.

We study dynamic network flows with uncertain input data under a robust optimization perspective. In the dynamic maximum flow problem, the goal is to maximize the flow reaching the sink within a given time horizon T , while flow requires a certain travel time to traverse an edge.In our setting, we account for uncertain travel times of flow. We investigate maximum flows over time under the assumption that at most Γ travel times may be prolonged simultaneously due to delay. We develop and study a mathematical model for this problem. As the dynamic robust flow problem generalizes the static version, it is NP-hard to compute an optimal flow. However, our dynamic version is considerably more complex than the static version. We show that it is NP-hard to verify feasibility of a given candidate solution. Furthermore, we investigate temporally repeated flows and show that in contrast to the non-robust case (that is, without uncertainties) they no longer provide optimal solutions for the robust problem, but rather yield a worst case optimality gap of at least T . We finally show that the optimality gap is at most O(ηk log T ), where η and k are newly introduced instance characteristics and provide a matching lower bound instance with optimality gap Ω(log T ) and η = k = 1. The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times.

In cost sharing games, the existence and efficiency of pure Nash equilibria fundamentally depends on the method that is used to share the resources' costs. We consider a general class of resource allocation problems in which a set of resources is used by a heterogeneous set of selfish users. The cost of a resource is a (non-decreasing) function of the set of its users. Under the assumption that the costs of the resources are shared by uniform cost sharing protocols, i.e., protocols that use only local information of the resource's cost structure and its users to determine the cost shares, we exactly quantify the inefficiency of the resulting pure Nash equilibria. Specifically, we show tight bounds on prices of stability and anarchy for games with only submodular and only supermodular cost functions, respectively, and an asymptotically tight bound for games with arbitrary set-functions. While all our upper bounds are attained for the well-known Shapley cost sharing protocol, our lower bounds hold for arbitrary uniform cost sharing protocols and are even valid for games with anonymous costs, i.e., games in which the cost of each resource only depends on the cardinality of the set of its users.

Stabilization of graphs has received substantial attention in recent years due to its connection to game theory. Stable graphs are exactly the graphs inducing a matching game with non-empty core. They are also the graphs that induce a network bargaining game with a balanced solution. A graph with weighted edges is called stable if the maximum weight of an integral matching equals the cost of a minimum fractional weighted vertex cover. If a graph is not stable, it can be stabilized in different ways. Recent papers have considered the deletion or addition of edges and vertices in order to stabilize a graph. In this work, we focus on a fine-grained stabilization strategy, namely stabilization of graphs by fractionally increasing edge weights.We show the following results for stabilization by minimum weight increase in edge weights (min additive stabilizer): (i) Any approximation algorithm for min additive stabilizer that achieves a factor of O(|V | 1/24−ǫ ) for ǫ > 0 would lead to improvements in the approximability of densestk-subgraph. (ii) Min additive stabilizer has no o(log |V |) approximation unless NP=P. Results (i) and (ii) together provide the first super-constant hardness results for any graph stabilization problem. On the algorithmic side, we present (iii) an algorithm to solve min additive stabilizer in factorcritical graphs exactly in poly-time, (iv) an algorithm to solve min additive stabilizer in arbitrary-graphs exactly in time exponential in the size of the Tutte set, and (v) a poly-time algorithm with approximation factor at most |V | for a super-class of the instances generated in our hardness proofs.

We study online resource allocation problems with a diseconomy of scale. In these problems, there are certain requests, each demanding a set of resources, that arrive in an online manner. The cost of each resource is semi-convex and grows superlinearly in the total load on the resource. An irrevocable allocation decision has to be made directly after the arrival of each request with the goal to minimize the total cost on the resources. We focus on two simple greedy online policies that provide very fast and easy approximation algorithms.The first policy is to minimize the individual cost of the current online request with respect to all previous requests that have been allocated before. The second policy is to minimize the marginal total cost over all requests that have arrived up to this point. In the literature, these type of algorithms is also considered as one-round walks in congestion games starting from the empty state.We consider the weighted and unweighted version of the problem. In the weighted variant, and for cost functions that are polynomials with maximal degree d and positive coefficients, we proof a tight competitive ratio of d √ 2 − 1 −(d+1) for the marginal total cost policy. This interestingly exactly matches the approximation factor for the corresponding multiple-round walk algorithm. Our work indicates that one-round walks that start in an empty starting state are exactly as efficient as multiple-round walks. We also show that this does not carry over to the unweighted version of the problem. For unweighted instances, we provide lower bounds for both policies that are significantly larger than the corresponding multiple-round walks. We complement our results with an upper and lower bound on the solution quality of the personal cost policy for weighted and unweighted instances.