Computational Complexity of Envy-free and Exchange-stable Seat Arrangement Problems on Grid Graphs

The Seat Arrangement Problem is a problem of finding a desirable seat arrangement for given preferences of agents and a seat graph that represents a configuration of seats. In this paper, we consider decision problems of determining if an envy-free arrangement exists and an exchange-stable arrangement exists, when a seat graph is an $\ell \times m$ grid graph. When $\ell=1$, the seat graph is a path of length $m$ and both problems have been known to be NP-complete. In this paper, we extend it and show that both problems are NP-complete for any integer $\ell \geq 2$.