6 research outputs found
Deciding the Winner of an Arbitrary Finite Poset Game is PSPACE-Complete
A poset game is a two-player game played over a partially ordered set (poset)
in which the players alternate choosing an element of the poset, removing it
and all elements greater than it. The first player unable to select an element
of the poset loses. Polynomial time algorithms exist for certain restricted
classes of poset games, such as the game of Nim. However, until recently the
complexity of arbitrary finite poset games was only known to exist somewhere
between NC^1 and PSPACE. We resolve this discrepancy by showing that deciding
the winner of an arbitrary finite poset game is PSPACE-complete. To this end,
we give an explicit reduction from Node Kayles, a PSPACE-complete game in which
players vie to chose an independent set in a graph
Strategy-Stealing Is Non-Constructive
In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature
The Computational Complexity of the Game of Set and its Theoretical Applications
The game of SET is a popular card game in which the objective is to form Sets
using cards from a special deck. In this paper we study single- and multi-round
variations of this game from the computational complexity point of view and
establish interesting connections with other classical computational problems.
Specifically, we first show that a natural generalization of the problem of
finding a single Set, parameterized by the size of the sought Set is W-hard;
our reduction applies also to a natural parameterization of Perfect
Multi-Dimensional Matching, a result which may be of independent interest.
Second, we observe that a version of the game where one seeks to find the
largest possible number of disjoint Sets from a given set of cards is a special
case of 3-Set Packing; we establish that this restriction remains NP-complete.
Similarly, the version where one seeks to find the smallest number of disjoint
Sets that overlap all possible Sets is shown to be NP-complete, through a close
connection to the Independent Edge Dominating Set problem. Finally, we study a
2-player version of the game, for which we show a close connection to Arc
Kayles, as well as fixed-parameter tractability when parameterized by the
number of rounds played
The Ungar Games
Let be a finite lattice. An Ungar move sends an element to the
meet of , where is a subset of the set of elements covered by
. We introduce the following Ungar game. Starting at the top element of ,
two players -- Atniss and Eeta -- take turns making nontrivial Ungar moves; the
first player who cannot do so loses the game. Atniss plays first. We say is
an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a
winning strategy in the Ungar game on . We first prove that the number of
principal order ideals in the weak order on that are Eeta wins is
. We then consider a broad class of intervals in Young's
lattice that includes all principal order ideals, and we characterize the Eeta
wins in this class; we deduce precise enumerative results concerning order
ideals in rectangles and type- root posets. We also characterize and
enumerate principal order ideals in Tamari lattices that are Eeta wins.
Finally, we conclude with some open problems and a short discussion of the
computational complexity of Ungar games.Comment: 23 pages, 6 figure
The Computational Complexity of Some Games and Puzzles With Theoretical Applications
The subject of this thesis is the algorithmic properties of one- and two-player
games people enjoy playing, such as Sudoku or Chess. Questions asked about puzzles
and games in this context are of the following type: can we design efficient computer
programs that play optimally given any opponent (for a two-player game), or solve
any instance of the puzzle in question?
We examine four games and puzzles and show algorithmic as well as intractability
results. First, we study the wolf-goat-cabbage puzzle, where a man wants to transport
a wolf, a goat, and a cabbage across a river by using a boat that can carry only one
item at a time, making sure that no incompatible items are left alone together. We
study generalizations of this puzzle, showing a close connection with the Vertex
Cover problem that implies NP-hardness as well as inapproximability results.
Second, we study the SET game, a card game where the objective is to form
sets of cards that match in a certain sense using cards from a special deck. We
study single- and multi-round variations of this game and establish interesting con-
nections with other classical computational problems, such as Perfect Multi-
Dimensional Matching, Set Packing, Independent Edge Dominating Set,
and Arc Kayles. We prove algorithmic and hardness results in the classical and
the parameterized sense.
Third, we study the UNO game, a game of colored numbered cards where players
take turns discarding cards that match either in color or in number. We extend results
by Demaine et. al. (2010 and 2014) that connected one- and two-player generaliza-
tions of the game to Edge Hamiltonian Path and Generalized Geography,
proving that a solitaire version parameterized by the number of colors is fixed param-
eter tractable and that a k-player generalization for k greater or equal to 3 is PSPACE-hard.
Finally, we study the Scrabble game, a word game where players are trying to
form words in a crossword fashion by placing letter tiles on a grid board. We prove
that a generalized version of Scrabble is PSPACE-hard, answering a question posed
by Demaine and Hearn in 2008