Lemmings Is PSPACE-Complete

Abstract

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 Forišek 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.

Gaming Is a Hard Job, but Someone Has to Do It!

“…To beat a level, at least a given percentage of lemmings has to reach one of several exit portals within a time limit. The complete set of rules, especially the ways lemmings behave in different landscapes, is quite complex, and has been described by the author in [14].…”

“…The NP-hardness of Lemmings was already proved by Cormode in [2] using only Digger skills. More recently, in [14], the author showed that Lemmings is PSPACE-complete, even if there is only one lemming in the level, and only Basher and Builder skills are available (the technique used is based on a variant of Metatheorem 4.c). Here we propose a simple alternative proof of NPhardness, which uses only Basher skills (that allow lemmings to dig horizontally) and relies on Metatheorem 3.b.…”

The Computational Complexity of Fire Emblem Series and similar Tactical Role-Playing Games

“…This section proves Theorem 1: Simplified FE is PSPACE-complete. We use the technique introduced in [30] and later used in [31,7,17], which is called the "open-close door" framework:…”

“…Many video games are proven to be NP-hard or PSPACE-complete [24,20,8,15,9,21,19,18,23,30,32,31,16,17,26,28,6]. Most of the games are puzzle-solving or path-finding types.…”

Quantified Boolean Formulas: Call the Plumber!

“…For example the computational complexity of Minesweeper [6], Sokoban [3], various Nintendo games, (for example Super Mario Bros) [1,4], Candy Crush [11,12], and Lemmings [10] has been analysed. To this avail in recent years several schemes (or frameworks) for the analysis of the computational complexity of famous video games have been proposed.…”