Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies -- that is, a centrally planned routing that minimizes the average cost of agents -- is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.

翻译：原子拥塞博弈是网络设计、路由和算法博弈论中的经典主题，能够模拟多种应用领域中的拥塞和流量优化任务。尽管此类博弈的无政府状态代价以及计算其纳什均衡的计算复杂性目前已得到充分理解，但文献中对计算系统最优策略集（即最小化代理平均成本的中心化规划路由）的计算复杂性研究严重不足。我们通过参数化复杂性范式的视角，确定了该问题可处理性的精确边界，从而填补了这一空白。在证明即使是在极简单的网络上该问题仍高度难解之后，我们获得的一系列结果表明，控制该问题计算（不）可处理性的结构参数本质上并非基于顶点分隔符（如树宽），而是基于边分隔符。最后，我们将分析扩展到该问题的（更具挑战性的）最小-最大变体。