18 research outputs found
Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard
The twentieth century has seen the rise of a new type of video games targeted
at a mass audience of "casual" gamers. Many of these games require the player
to swap items in order to form matches of three and are collectively known as
\emph{tile-matching match-three games}. Among these, the most influential one
is arguably \emph{Bejeweled} in which the matched items (gems) pop and the
above gems fall in their place. Bejeweled has been ported to many different
platforms and influenced an incredible number of similar games. Very recently
one of them, named \emph{Candy Crush Saga} enjoyed a huge popularity and
quickly went viral on social networks. We generalize this kind of games by only
parameterizing the size of the board, while all the other elements (such as the
rules or the number of gems) remain unchanged. Then, we prove that answering
many natural questions regarding such games is actually \NP-Hard. These
questions include determining if the player can reach a certain score, play for
a certain number of turns, and others. We also
\href{http://candycrush.isnphard.com}{provide} a playable web-based
implementation of our reduction.Comment: 21 pages, 12 figure
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 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.Comment: 26 pages, 11 figure
Restricted Power - Computational Complexity Results for Strategic Defense Games
We study the game Greedy Spiders, a two-player strategic defense game, on planar graphs and show PSPACE-completeness for the problem of deciding whether one player has a winning strategy for a given instance of the game. We also generalize our results in metatheorems, which consider a large set of strategic defense games. We achieve more detailed complexity results by restricting the possible strategies of one of the players, which leads us to Sigma^p_2- and Pi^p_2-hardness results
The Computational Complexity of Portal and Other 3D Video Games
We classify the computational complexity of the popular video games Portal and Portal 2. We isolate individual mechanics of the game and prove NP-hardness, PSPACE-completeness, or pseudo-polynomiality depending on the specific game mechanics allowed. One of our proofs generalizes to prove NP-hardness of many other video games such as Half-Life 2, Halo, Doom, Elder Scrolls, Fallout, Grand Theft Auto, Left 4 Dead, Mass Effect, Deus Ex, Metal Gear Solid, and Resident Evil. These results build on the established literature on the complexity of video games [Aloupis et al., 2014][Cormode, 2004][Forisek, 2010][Viglietta, 2014]
Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets
A door gadget has two states and three tunnels that can be traversed by an
agent (player, robot, etc.): the "open" and "close" tunnel sets the gadget's
state to open and closed, respectively, while the "traverse" tunnel can be
traversed if and only if the door is in the open state. We prove that it is
PSPACE-complete to decide whether an agent can move from one location to
another through a planar assembly of such door gadgets, removing the
traditional need for crossover gadgets and thereby simplifying past
PSPACE-hardness proofs of Lemmings and Nintendo games Super Mario Bros., Legend
of Zelda, and Donkey Kong Country. Our result holds in all but one of the
possible local planar embedding of the open, close, and traverse tunnels within
a door gadget; in the one remaining case, we prove NP-hardness.
We also introduce and analyze a simpler type of door gadget, called the
self-closing door. This gadget has two states and only two tunnels, similar to
the "open" and "traverse" tunnels of doors, except that traversing the traverse
tunnel also closes the door. In a variant called the symmetric self-closing
door, the "open" tunnel can be traversed if and only if the door is closed. We
prove that it is PSPACE-complete to decide whether an agent can move from one
location to another through a planar assembly of either type of self-closing
door. Then we apply this framework to prove new PSPACE-hardness results for
eight different 3D Mario games and Sokobond.Comment: Accepted to FUN2020, 35 pages, 41 figure