Lemmings is PSPACE-complete
Lemmings is a computer puzzle game developed by DMA Design and published by Psygnosis in 1991, in which the player has to guide a tribe of lemming creatures to safety through a hazardous landscape, by assigning them specific skills that modify their behavior in different ways. In this paper we study the optimization problem of saving the highest number of lemmings in a given landscape with a given number of available skills. We prove that, if the number of Builder skills is bounded by a polynomial in the size of the landscape, then the maximization problem belongs to NPO. We also show that, with no such restriction, the game is PSPACE-complete, even if there is only one lemming to save. We thereby settle an open problem posed by Cormode in 2004, and again by Forišek in 2010. Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only Climber skills are available. This contrasts with the membership in PO of the same optimization problem restricted to landscapes with no " deadly areas " (such as water or traps) and only Climber and Floater skills, as previously established by Cormode.