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

TL;DR

This work introduces Crossing Cost Multi-Agent Routing (CC-MAR), a CC-based congestion-game model on mixed graphs for asynchronous, decentralized routing where head-on interactions are captured by edge crossing costs rather than hard collision constraints. It establishes a robust theoretical foundation: pure Nash equilibria always exist with convergent best-response dynamics, the Price of Stability is 1, while the Price of Anarchy can be unbounded; computing a Nash equilibrium is poly-time under polynomially-bounded weights and PLS-complete in general. On the optimization side, minimizing the total crossing cost is NP-hard (generalizes Steiner Orientation), motivating a comprehensive parameterized-complexity framework that yields XP and multiple FPT results across parameters such as the number of terminal pairs, edges, arcs, diameter, and vertex cover, with tightness results linked to Steiner Orientation. The work situates CC-MAR at the intersection of MAPF, congestion games, and graph orientation, offering a theoretical basis for scalable, decentralized routing in networks where synchronization is impractical and risks must be managed via crossing costs. Overall, the paper provides both foundational equilibrium guarantees and practical, parameterized algorithms to guide decentralized, risk-aware routing in complex networks.

Abstract

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.

Figures (5)

• Figure 1: An example of CC-MAR. In this example, there are 4 agents with terminal pairs $(v_{11},v_{15})$, $(v_{35},v_{12})$, $(v_{31},v_{35})$, and $(v_{33},v_{34})$, respectively. The strategy of each agent is $P_i$ for $i\in \{1,2,3,4\}$. Path $P_1$ and path $P_2$ cross each other on the edge $\{v_{12},v_{13}\}$. Thus, the crossing cost between $P_1$ and $P_2$ is 1. The individual cost of agent 1 is also 1, as it has no other crossings. On the other hand, the individual cost of agent 2 is 4. This is because it crosses $P_1$ on edge $\{v_{12},v_{13}\}$, crosses $P_3$ on edges $\{v_{33},v_{34}\},\{v_{34},v_{35}\}$, and crosses $P_4$ on edge $\{v_{34},v_{35}\}$. The total cost of the strategy profile is the sum of all pairwise crossing costs, which is $1 (\text{from } P_1, P_2) + 2 (\text{from } P_2, P_3) + 1 (\text{from } P_2, P_4) = 4$.
• Figure 2: A grid-like graph when $n=4$. The blue arcs are threshold arcs.
• Figure 3: The replacement of arc $(u,v)$ by a mixed $u$-$v$ path consisting of $(u,u'), \{u',v'\},(v',v)$.
• Figure 4: The resource gadget of $r_{ij}$ obtained by replacing vertex $v_{ij}$.
• Figure 5: Case analysis in Claim \ref{['claim:onepath:cross']}.

Theorems & Definitions (33)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 1
• Theorem 5