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 prove a general monotonicity result about Nash flows in directed networks and use it for the design of truthful mechanisms in the setting where each edge of the network is controlled by a different selfish agent, who incurs costs when her edge is used. The costs for each edge are assumed to be linear in the load on the edge. To compensate for these costs, the agents impose tolls for the usage of edges. When nonatomic selfish network users choose their paths through the network independently and each user tries to minimize a weighted sum of her latency and the toll she has to pay to the edges, a Nash flow is obtained. Our monotonicity result implies that the load on an edge in this setting can not increase when the toll on the edge is increased, so the assignment of load to the edges by a Nash flow yields a monotone algorithm. By a well-known result, the monotonicity of the algorithm then allows us to design truthful mechanisms based on the load assignment by Nash flows. Moreover, we consider a mechanism design setting with two-parameter agents, which is a generalization of the case of one-parameter agents considered in a seminal paper of Archer and Tardos. While the private data of an agent in the one-parameter case consists of a single nonnegative real number specifying the agent's cost per unit of load assigned to her, the private data of a two-parameter agent consists of a pair of nonnegative real numbers, where the first one specifies the cost of the agent per unit load as in the one-parameter case, and the second one specifies a fixed cost, which the agent incurs independently of the load assignment. We give a complete characterization of the set of output functions that can be turned into truthful mechanisms for two-parameter agents. Namely, we prove that an output function for the two-parameter setting can be turned into a truthful mechanism if and only if the load assigned to every agent is nonincreasing in the agent's bid for her per unit cost and, for almost all fixed bids for the agent's per unit cost, the load assigned to her is independent of the agent's bid for her fixed cost. When the load assigned to an agent is continuous in the agent's bid for her per unit cost, it must be completely independent of the agent's bid for her fixed cost. These results motivate our choice of linear cost functions without fixed costs for the edges in the selfish routing setting, but the results also seem to be interesting in the context of algorithmic mechanism design themselves.
In this paper we introduce a cooperative game based on the minimum cut tree problem which is also known as multi-terminal maximum flow problem. Minimum cut tree games are shown to be totally balanced and a solution in their core can be obtained in polynomial time. This special core allocation is closely related to the solution of the original graph theoretical problem. We give an example showing that the game is not supermodular in general, however, it is for special cases and for some of those we give an explicit formula for the calculation of the Shapley value.
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.
In this paper we consider the location of stops along the edges of an already existing public transportation network, as introduced in [SHLW02]. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The goal is to achieve a maximal covering of given demand points with a minimal number of stops. This bicriterial problem is in general NP-hard. We present a nite dominating set yielding an IP-formulation as a bicriterial set covering problem. We use this formulation to observe that along one single straight line the bicriterial stop location problem can be solved in polynomial time and present an e cient solution approach for this case. It can be used as the basis of an algorithm tackling real-world instances.