Friends-and-strangers is PSPACE-complete

Abstract

The friends-and-strangers problem is defined on two graphs X and Y of the same order n, where X is the location graph on a set of locations $V\left(X\right)$ and Y is the friendship graph on a set of people $V\left(Y\right)$. A configuration is a bijection from $V\left(X\right)$ to $V\left(Y\right)$, representing an arrangement of n people at n locations. The friends-and-stranger problem investigates the connectivity of two arbitrary configurations by a sequence of swaps of two people that are neighbors (i.e., adjacent in X) and friends (i.e., adjacent in Y). In this paper, we show that the friends-and-strangers problem with subcubic planar location graphs X is PSPACE-complete by a reduction from the Ncl (non-deterministic constraint logic) problem.