Computational Complexity of Envy-free and Exchange-stable Seat Arrangement Problems on Grid Graphs
Sota Kawase, Shuichi Miyazaki
TL;DR
The paper addresses the hardness of finding fair seat arrangements under envy-free and exchange-stable criteria on grid-based seat graphs, proving NP-completeness for every grid with $\\ell \ge 2$ via reductions from the Hamiltonian path problem and its variant DHP*. It develops gadget-based reductions to simulate Hamiltonian-path constraints within grid layouts and to align envy relations with exchange-stability, handling cases from $2 \times m$ up to $\\ell \times m$ grids. The key contributions are the NP-completeness results for both EFA and ESA on all grid sizes considered, extending prior path-specific hardness to multi-row grids, and highlighting the intricate grid-specific challenges in ensuring equivalence to Hamiltonian structure. These results establish fundamental hardness barriers for fair seat allocation in common seating configurations and guide future work toward identifying tractable special cases or restricted preference structures with polynomial-time solutions.
Abstract
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 $\ell \geq 2$.