Optimal Seat Arrangement: What Are the Hard and Easy Cases?

We study four NP-hard optimal seat arrangement problems which each have as input a set of n agents, where each agent has cardinal preferences over other agents, and an n-vertex undirected graph (called the seat graph). The task is to assign each agent to a distinct vertex in the seat graph such that either the sum of utilities or the minimum utility is maximized, or it is envy-free or exchange-stable. Aiming at identifying hard and easy cases, we extensively study the algorithmic complexity of the four problems by looking into natural graph classes for the seat graph (e.g., paths, cycles, stars, or matchings), problem-specific parameters (e.g., the number of non-isolated vertices in the seat graph or the maximum number of agents towards whom an agent has non-zero preferences), and preference structures (e.g., non-negative or symmetric preferences). For strict preferences and seat graphs with disjoint edges and isolated vertices, we correct an error in the literature and show that finding an envy-free arrangement remains NP-hard in this case.