Coordinating the movement of multiple autonomous agents over a shared network is a fundamental challenge in algorithmic robotics, intelligent transportation, and distributed systems. The dominant approach, Multi-Agent Path Finding, relies on centralized control and synchronous collision avoidance, which often requires strict synchronization and guarantees of globally conflict-free execution. This paper introduces the Multi-Agent Routing under Crossing Cost model on mixed graphs, a novel framework tailored to asynchronous settings. In our model, instead of treating conflicts as hard constraints, each agent is assigned a path, and the system is evaluated through a cost function that measures potential head-on encounters. This ``crossing cost'', which is defined as the product of the numbers of agents traversing an edge in opposite directions, quantifies the risk of congestion and delay in decentralized execution. Our contributions are both game-theoretic and algorithmic. We model the setting as a congestion game with a non-standard cost function, prove the existence of pure Nash equilibria, and analyze the dynamics leading to them. Equilibria can be found in polynomial time under mild conditions, while the general case is PLS-complete. From an optimization perspective, minimizing the total crossing cost is NP-hard, as the problem generalizes Steiner Orientation. To address this hardness barrier, we design a suite of parameterized algorithms for minimizing crossing cost, with parameters including the number of arcs, edges, agents, and structural graph measures. These yield XP or FPT results depending on the parameter, offering algorithmic strategies for structurally restricted instances. Our framework provides a new theoretical foundation for decentralized multi-agent routing, bridging equilibrium analysis and parameterized complexity to support scalable and risk-aware coordination.

翻译：在共享网络上协调多个自主智能体的移动是算法机器人学、智能交通和分布式系统中的基本挑战。主流方法——多智能体路径规划依赖于集中式控制和同步避碰，通常需要严格的同步性并保证全局无冲突执行。本文针对异步场景，提出了一种在混合图上的交叉成本下多智能体路由模型。在该模型中，冲突不再被视为硬约束，而是为每个智能体分配一条路径，并通过衡量潜在对向相遇的成本函数来评估系统性能。这种“交叉成本”定义为在边上沿相反方向通行的智能体数量的乘积，它量化了去中心化执行中拥堵和延迟的风险。我们的贡献兼具博弈论与算法性质：将场景建模为具有非标准成本函数的拥塞博弈，证明纯纳什均衡的存在性，并分析导致均衡的动态过程。在温和条件下可在多项式时间内找到均衡，而一般情况则属于PLS完全问题。从优化视角看，最小化总交叉成本是NP难的，因为该问题推广了斯坦纳定向问题。为应对这一计算困难，我们设计了一套用于最小化交叉成本的参数化算法，参数包括弧数、边数、智能体数量以及图的结构度量。根据参数的不同，这些算法可产生XP或FPT结果，为结构受限的实例提供了算法策略。本框架为去中心化多智能体路由奠定了新的理论基础，通过连接均衡分析与参数化复杂度，为可扩展且风险感知的协调机制提供了支撑。