Game-Theoretic and Algorithmic Analyses of Multi-Agent Routing under Crossing Costs

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.