We study the existence of pure Nash equilibria in weighted congestion games. Let denote a set of cost functions. We say that is consistent if every weighted congestion game with cost functions in possesses a pure Nash equilibrium. Our main contribution is a complete characterization of consistency of continuous cost functions. We prove that a set of continuous functions is consistent for two-player games if and only if contains only monotonic functions and for all nonconstant functions c 1 c 2 ∈ , there are constants a b ∈ such that c 1 x = a c 2 x + b for all x ∈ ≥0 . For games with at least three players, we prove that is consistent if and only if exactly one of the following cases holds: (a) contains only affine functions; (b) contains only exponential functions such that c x = a c e x + b c for some a c b c ∈ , where a c and b c may depend on c, while must be equal for every c ∈ . The latter characterization is even valid for three-player games. Finally, we derive several characterizations of consistency of cost functions for games with restricted strategy spaces, such as weighted network congestion games or weighted congestion games with singleton strategies.

We study dynamic network flows and introduce a notion of instantaneous dynamic equilibrium (IDE) requiring that for any positive inflow into an edge, this edge must lie on a currently shortest path towards the respective sink. We measure current shortest path length by current waiting times in queues plus physical travel times. As our main results, we show: 1. existence and constructive computation of IDE flows for multi-source single-sink networks assuming constant network inflow rates, 2. finite termination of IDE flows for multi-source single-sink networks assuming bounded and finitely lasting inflow rates, 3. the existence of IDE flows for multi-source multi-sink instances assuming general measurable network inflow rates, 4. the existence of a complex single-source multi-sink instance in which any IDE flow is caught in cycles and flow remains forever in the network.

Joint use of resources with usage-dependent cost raises the question: who pays how much? We study cost sharing in resource selection games where the strategy spaces are either singletons or bases of a matroid defined on the ground set of resources. Our goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate three classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium; separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource; uniform protocols are separable and require that the cost shares on a resource may not depend on the instance, that is, they remain the same even if new resources are added to or removed from the instance. We find optimal basic and separable protocols that guarantee the price of stability and price of anarchy to grow logarithmically in the number of players, except for the case of matroid games induced by separable protocols where the price of anarchy grows linearly with the number of players. For uniform protocols we show that the price of anarchy is unbounded even for singleton games.

An approximation algorithm for an optimization problem runs in polynomial time for all instances and is guaranteed to deliver solutions with bounded optimality gap. We derive such algorithms for different variants of capacitated location routing, an important generalization of vehicle routing where the cost of opening the depots from which vehicles operate is taken into account. Our results originate from combining algorithms and lower bounds for different relaxations of the original problem; along with location routing we also obtain approximation algorithms for multidepot capacitated vehicle routing by this framework. Moreover, we extend our results to further generalizations of both problems, including a prize-collecting variant, a group version, and a variant where cross-docking is allowed. We finally present a computational study of our approximation algorithm for capacitated location routing on benchmark instances and large-scale randomly generated instances. Our study reveals that the quality of the computed solutions is much closer to optimality than the provable approximation factor.