Gaming is a hard job, but someone has to do it!
Giovanni Viglietta
University of Pisa, Italy,
viglietta@gmail.com
Abstract. We establish some general schemes relating the computa-
tional complexity of a video game to the presence of certain common ele-
ments or mechanics, such as destroyable paths, collecting items, doors ac-
tivated by switches or pressure plates, etc.. Then we apply such \metathe-
orems" to several video games published between 1980 and 1998, includ-
ing Pac-Man, Tron, Lode Runner, Boulder Dash, De
ektor, Mindbender,
Pipe Mania, Skweek, Prince of Persia, Lemmings, Doom, Puzzle Bob-
ble 3, and Starcraft. We obtain both new results, and improvements or
alternative proofs of previously known results.
1 Introduction
This work was inspired mainly by the recent papers on the computational
complexity of video games by Forisek [4] and Cormode [2], along with the
excellent surveys on the topic by Kendall et al. [6] and Demaine et al. [3],
and may be regarded as their continuation on the same line of research.
Our purpose is to single out certain recurring features or mechanics
in a video game that enable general reduction schemes from known hard
problems to the games we are considering. To this end, in Section 2 we pro-
duce several metatheorems that will be applied in Section 3 to a wealth
of famous commercial video games, in order to automatically establish
their hardness with respect to certain computational complexity classes
(with a couple of exceptions). Because most recent commercial games
incorporate Turing-equivalent scripting languages that easily allow to de-
sign undecidable puzzles as part of the gameplay, we will focus primarily
on older, \scriptless" games. Our selection includes games published be-
tween 1980 and 1998, presented in alphabetical order for better reference.
Due to space limitations, not every game is properly introduced, but our
constructions should be promptly understood by any casual player.
Several open problems remain: Whenever only the hardness of a game
is proved with respect to some complexity class, the obviously implied
question is whether the game is also complete for that class. Dierent
variants of each game may be studied, obtained for instance by further
restricting the set of game elements used in our hardness proofs.
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The reader is assumed to be familiar with general computational com-
plexity theoretic concepts and classes: For an introduction, refer to [9].
2 Metatheorems
More often than not, games allow the player to control an avatar, either
directly or indirectly. In some circumstances, an avatar may be identied
within the game only through some sort of artice or abstraction on the
game mechanics. Throughout Section 2, we will stipulate that the player's
actions involve controlling an avatar, and that the elements of the game
may be freely arranged in a plane lattice, or a higher dimensional space.
2.1 Location traversal and single-use paths
A game is said to exhibit the location traversal feature if the level de-
signer can somehow force the player's avatar to visit several specic game
locations, arbitrarily connected together, in order to beat the level. Lo-
cations may be visited multiple times in any order, but the rst one is
usually xed (starting location), and sometimes also the last one is (exit
location). An example of location traversal feature is the collecting items
feature discussed in [4]: A certain number of items are scattered across
dierent locations, and the avatar's task is to collect them all.
The single-use paths feature is the existence of congurations of game
elements that act as paths connecting two locations, which can be tra-
versed by the avatar at most once.
Metatheorem 1. Any game exhibiting both location traversal and single-
use paths is NP-hard.
Proof. We give a straightforward reduction from Hamiltonian Cycle,
which is NP-complete even for 3-regular planar graphs. Construct a plane
embedding of the given 3-regular graph (perhaps an orthogonal embed-
ding, if needed) with an additional node u dangling from a distinguished
node v. Nodes are locations that must be visited, and edges are imple-
mented as single-use paths. The starting location is placed in v and, if an
exit location is required, it is placed in u. ut
As Section 3 testies, Metatheorem 1 has a wide range of applications,
and it tends to yield game levels that are more \playable" than those
resulting from the somewhat analogous [4, Metathm. 2], which rely on a
tight time limit to traverse a grid graph. Additionally, [4, Metathm. 2]
is prone to design complications in games where the avatar moves at
dierent speeds in dierent directions, for instance due to gravity eects.