排除Braess悖论并保持随时间流单调性的网络拓扑
Network Topology That Excludes Braess's Paradox and Maintains Monotonicity in Flows Over Time
- SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文刻画了随时间流博弈中排除Braess悖论的网络拓扑（双极伪树状链），并证明其满足流入速率与完工时间的单调性，解决了两个开放猜想。
AI中文摘要：
在“随时间流”博弈中，无穷小的流动粒子旨在尽可能快地通过网络从源点到达汇点。在Vickrey瓶颈模型下，网络边上的拥塞效应通过FIFO队列来刻画，当边的流入量超过其容量时就会出现这种队列。本文针对该博弈的两个开放猜想展开研究：1) 对Braess悖论免疫的网络的刻画；2) 网络流入速率与整体流完工时间之间的单调性关系。我们证明，一类被称为“双极伪树状链”（BPAs）的网络拓扑完全解决了第一个猜想，并在第二个猜想上取得了部分进展。BPAs链恰好构成了Macko等人（2013）在研究随时间流的Braess悖论时留下的未解情形。通过结构化刻画此类网络上Nash流随时间的动态演化，我们证明了从这些网络中移除边永远不会降低最大均衡延迟。结合Macko等人的结果，这建立了一个充要条件：一个网络不出现随时间流的Braess悖论当且仅当它是BPAs链，从而解决了他们的猜想。此外，我们确认了Correa等人（2021）对所有BPAs链的单调性猜想。该结果严格推广了先前在均匀流入下平行路径链的已知单调性。
英文摘要：
In the game of \emph{flow over time}, infinitesimal flow particles aim to travel from a source to a sink in a network as quickly as possible. Under the Vickrey bottleneck model, the congestion effects on network edges are captured through FIFO queues, which arise when the inflow into an edge exceeds its capacity. This work addresses two open conjectures about the game: 1) the characterization of networks that are immune to Braess's paradox, and 2) the monotonicity relationship between the network inflow rate and the overall flow makespan. We show that a single class of network topologies, called \emph{chains of bipolar pseudo-arborescences} (\emph{BPAs}), fully resolves the first conjecture and yields partial progress on the second. Chains of BPAs constitute precisely the cases left open by Macko et al.~(2013) in their study of Braess's paradox for flow over time. By structurally characterizing the dynamic evolution of Nash flows over time on such networks, we prove that removing edges from these networks never decreases the maximum equilibrium latency. Combined with the results of Macko et al., this establishes a necessary and sufficient condition: \emph{a network does not admit Braess's paradox for flow over time if and only if it is a chain of BPAs}, thereby resolving their conjecture. Furthermore, we confirm the monotonicity conjecture of Correa et al.~(2021) for all chains of BPAs. This result strictly generalizes the previously known monotonicity for chains of parallel paths under uniform inflows.
