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.

中文翻译：

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优化原子拥塞的复杂性


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，能够对各种应用领域的拥塞和流量优化任务进行建模。虽然此类游戏的无政府状态的代价以及计算其纳什均衡的计算复杂性现在已经很好理解，但计算一组系统最优策略的计算复杂性——即，最小化代理平均成本的集中规划路由——在文献中研究严重不足。我们通过参数化复杂性范式的视角确定问题可处理性的确切边界来缩小这一差距。在证明即使在极其简单的网络上，问题仍然非常棘手之后，我们得到了一组结果，这些结果表明，控制问题计算（不）易处理性的结构参数本质上不是基于顶点分隔符的（例如，树宽），而是基于边缘分隔符。最后，我们将分析扩展到问题的（更具挑战性的）最小-最大变体。