This work analyzes the minimum tollbooth problem in atomic network congestion games with unsplittable flows. The goal is to place tolls on edges, such that there exists a pure Nash equilibrium in the tolled game that is a social optimum in the untolled one. Additionally, we require the number of tolled edges to be the minimum. This problem has been extensively studied in non-atomic games, however, to the best of our knowledge, it has not been considered for atomic games before. By a reduction from the weighted CNF SAT problem, we show both the NP-hardness of the problem and the W[2]-hardness when parameterizing the problem with the number of tolled edges. On the positive side, we present a polynomial time algorithm for networks on series-parallel graphs that turns any given state of the untolled game into a pure Nash equilibrium of the tolled game with the minimum number of tolled edges.

这项工作分析了流量不可分割的原子网络拥塞游戏中的最低收费站问题。目的是将通行费置于边缘，以便在收费游戏中存在纯纳什均衡，这在无收费游戏中是社会最优的。此外，我们要求收费边的数量必须最少。在非原子游戏中已经对该问题进行了广泛的研究，但是，据我们所知，以前从未在原子游戏中考虑过此问题。通过减少加权CNF SAT问题，我们同时显示了问题的NP硬度和W [2]参数化带通行费边缘数量的问题时的硬度。在积极方面，我们提出了一种用于网络的并行时间多项式时间算法，该算法将无收费游戏的任何给定状态转换为具有最小收费边数的收费游戏的纯Nash平衡。