Designing Pairwise-Stable Agent Seating Arrangements

Many fundamental problems in multi-agent systems involve the arrangement of agents, who have preferences over each other, on a target graph. These problems include, for example, Stable Matching, Seat Arrangement, and Coalition Formation. However, guaranteeing game-theoretically desirable properties such as exchange-stability or envy-freeness is difficult, as such solutions may not exist, and even if they do, they are often intractable to find, even in highly constrained settings such as path or cycle target graphs. In this paper, we challenge the classical setup and investigate what can be achieved when the structure of the target graph is a designable object for the central planner, rather than a fixed part of the input. We study this in the context of a natural pairwise stability criterion, which is similar to having spare seats. In particular, we introduce a highly flexible framework to efficiently design approximately optimal target graphs and associated pairwise-stable agent arrangements. Our model assumes that agents have (weak or strict) ordinal preferences over other agents. We show that classical results from stable matching theory can be extended and adapted to this much more general setting and can serve as a useful tool for navigating the trade-off between stability and computational efficiency. Our results highlight strict boundaries between tractability and intractability, and between local and global optimality. We also uncover intriguing connections to classical computational problems such as subgraph isomorphism, disjoint path partitioning, and bin-packing.