We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network... more We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NPhard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.
We present an exact algorithm for computing an earliest arrival flow in a discrete time setting o... more We present an exact algorithm for computing an earliest arrival flow in a discrete time setting on series-parallel graphs. In contrast to previous results for the earliest arrival flow problem this algorithm runs in polynomial time
We study the efficient computation of Nash and strong equi-libria in weighted bottleneck games. I... more We study the efficient computation of Nash and strong equi-libria in weighted bottleneck games. In such a game different players interact on a set of resources in the way that every player chooses a sub-set of the resources as her strategy. The cost of a single resource depends on the total weight of players choosing it and the personal cost every player tries to minimize is the cost of the most expensive resource in her strategy, the bottleneck value. To derive efficient algorithms for finding Nash equilibria in these games, we generalize a tranformation of a bottleneck game into a special con-gestion game introduced by Caragiannis et al. [1]. While investigating the transformation we introduce so-called lexicographic games, in which the aim of a player is not only to minimize her bottleneck value but to lexicographically minimize the ordered vector of costs of all resources in her strategy. For the special case of network bottleneck games, i.e., the set of resources are the edges ...
We consider the problem of routing traffic by selfish users in a net-work G = (V, E) with load-de... more We consider the problem of routing traffic by selfish users in a net-work G = (V, E) with load-dependent nondecreasing latency functions ℓe on the edges. Each network user routes its traffic on a path of minimum latency, given the latency on the edges caused by the other users. It is known that under standard assumptions the routes chosen by the users forms a Nash equi-librium which, in general, is different from the global optimum. Both, a Nash equilibrium and the optimum, can be computed in polynomial time by convex programming, provided the latency functions are also differentiable. We give simple polynomial algorithms which for a series-parallel graph G with affine latency functions ℓe(x) = aex + be explicitly construct as functions of the total traffic r to be routed (i) the common latency LG(r) experienced by the users in a Nash equilibrium, (ii) for each edge e ∈ E the flow fe(r) on e in a Nash equilibrium , and (iii) a Nash flow and a global optimum (minimum latency) flow.
We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion g... more We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it depends on the graph topology and the number of users. In our context best and worst equilibria are those with minimum respectively maximum total latency. We establish that both problems can be solved by a Greedy algorithm with a suitable tie breaking rule on parallel links. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. Additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.
We present an exact algorithm for computing an earliest arrival flow in a discrete time setting o... more We present an exact algorithm for computing an earliest arrival flow in a discrete time setting on series-parallel graphs. In contrast to previous results for the earliest arrival flow problem this algorithm runs in polynomial time.
We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network... more We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NPhard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.
We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network... more We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NPhard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.
We present an exact algorithm for computing an earliest arrival flow in a discrete time setting o... more We present an exact algorithm for computing an earliest arrival flow in a discrete time setting on series-parallel graphs. In contrast to previous results for the earliest arrival flow problem this algorithm runs in polynomial time
We study the efficient computation of Nash and strong equi-libria in weighted bottleneck games. I... more We study the efficient computation of Nash and strong equi-libria in weighted bottleneck games. In such a game different players interact on a set of resources in the way that every player chooses a sub-set of the resources as her strategy. The cost of a single resource depends on the total weight of players choosing it and the personal cost every player tries to minimize is the cost of the most expensive resource in her strategy, the bottleneck value. To derive efficient algorithms for finding Nash equilibria in these games, we generalize a tranformation of a bottleneck game into a special con-gestion game introduced by Caragiannis et al. [1]. While investigating the transformation we introduce so-called lexicographic games, in which the aim of a player is not only to minimize her bottleneck value but to lexicographically minimize the ordered vector of costs of all resources in her strategy. For the special case of network bottleneck games, i.e., the set of resources are the edges ...
We consider the problem of routing traffic by selfish users in a net-work G = (V, E) with load-de... more We consider the problem of routing traffic by selfish users in a net-work G = (V, E) with load-dependent nondecreasing latency functions ℓe on the edges. Each network user routes its traffic on a path of minimum latency, given the latency on the edges caused by the other users. It is known that under standard assumptions the routes chosen by the users forms a Nash equi-librium which, in general, is different from the global optimum. Both, a Nash equilibrium and the optimum, can be computed in polynomial time by convex programming, provided the latency functions are also differentiable. We give simple polynomial algorithms which for a series-parallel graph G with affine latency functions ℓe(x) = aex + be explicitly construct as functions of the total traffic r to be routed (i) the common latency LG(r) experienced by the users in a Nash equilibrium, (ii) for each edge e ∈ E the flow fe(r) on e in a Nash equilibrium , and (iii) a Nash flow and a global optimum (minimum latency) flow.
We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion g... more We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it depends on the graph topology and the number of users. In our context best and worst equilibria are those with minimum respectively maximum total latency. We establish that both problems can be solved by a Greedy algorithm with a suitable tie breaking rule on parallel links. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. Additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.
We present an exact algorithm for computing an earliest arrival flow in a discrete time setting o... more We present an exact algorithm for computing an earliest arrival flow in a discrete time setting on series-parallel graphs. In contrast to previous results for the earliest arrival flow problem this algorithm runs in polynomial time.
We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network... more We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NPhard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.
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Papers by Heike Sperber