Lemmings is PSPACE-complete

TL;DR

This paper proves that the puzzle game Lemmings is PSPACE-complete in general, but becomes NP when skill availability is polynomially bounded, and shows that maximizing saved lemmings is APX-hard.

Contribution

It settles the open problem of Lemmings' computational complexity and characterizes the complexity under various skill restrictions.

Findings

Lemmings is PSPACE-complete even with one lemming and limited skills.

The game is NP-complete when skills are polynomially bounded.

Maximizing saved lemmings is APX-hard, even with a single skill type.

Abstract

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 the game is PSPACE-complete, even if there is only one lemming to save, and only Builder and Basher skills are available. We thereby settle an open problem posed by Cormode in 2004, and again by Forisek in 2010. However we also prove that, if we restrict the game to levels in which the available Builder skills are only polynomially many (and there is any number of other skills), then the game is solvable in NP. Similarly, if the available Basher, Miner, and Digger skills are polynomially many, the game is solvable in NP. Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only one type of skill is available, whatever this skill is. This contrasts with the membership in P of the decision problem restricted to levels with no "deadly areas" (such as water or traps) and only Climber and Floater skills, as previously established by Cormode.