The Computational Complexity
of two Card Games with
Theoretical Applications
Valia Mitsou
Supported by the
Games and Graphs
Project
Today
Arc Kayles
Generalized Geography
Unbounded Games
●
●
Constant-size games are by definition trivially efficient.
We need to define unbounded versions of the games
we study.
n
n
Parameterized Games
➢
Games and Puzzles typically have several small
parameters (ex. # of suits in a card game).
➢
➢
Distinguish between truly hard games
and parameterized-efficient games.
Study short games.
It is White to move and
checkmate in 3 moves.
The Computational Complexity
of the Game of
Joint work with Michael Lampis
(Université Paris Dauphine)
The Game of Set - Rules
Each card has 4 attributes:
–
Symbol
–
Shading
–
Color
–
Number
The Game of Set - Rules
Each attribute can take one of 3 values:
–
Symbol
●
●
●
Oval
Diamond
Squiggle
The Game of Set - Rules
Each attribute can take one of 3 values:
–
Color
●
●
●
Red
Green
Purple
The Game of Set - Rules
Each attribute can take one of 3 values:
–
Shading
●
●
●
Blank
Stripped
Solid
The Game of Set - Rules
Each attribute can take one of 3 values:
–
Number
●
●
●
One
Two
Three
The Game of Set - Rules
Valid set: 3 cards with values for each attribute being
either all the same or all different.
✔
All have same color;
✔
all have same symbol;
✔
all have same number;
✔
all have different shadings.
The Game of Set - Rules
Valid set: 3 cards with values for each attribute being
either all the same or all different.
✔
All have different colors;
✔
all have different symbols;
✔
all have different numbers;
✔
all have same shading.
The Game of Set - Rules
This is not a valid set!
✔
All have same colors;
✔
all have different symbols;
×
only 2/3 have same number;
✔
all have different shadings.
A Turn-based SET Game
RULES:
●
●
Deal n cards.
Players take turns picking a valid set, removing
selected cards.
●
Game continues until there is no valid set left.
●
The last player to pick a valid set wins.
A Turn-based SET Game
RULES:
●
●
Deal n cards.
Players take turns picking a valid set, removing
selected cards.
●
Game continues until there is no valid set.
●
The last player to pick a valid set wins.
Player A
Player B
A Turn-based SET Game
RULES:
●
●
Deal n cards.
Players take turns picking a valid set, removing
selected cards.
●
Game continues until there is no valid set.
●
The last player to pick a valid set wins.
Player A
Player B
A Turn-based SET Game
RULES:
●
●
Deal n cards.
Players take turns picking a valid set, removing
selected cards.
●
Game continues until there is no valid set.
●
The last player to pick a valid set wins.
Player A
Player A wins!
Player B
Similar game: ARC KAYLES
ARC KAYLES: Played on a graph.
●
●
●
Players take turns picking edges from the graph.
Each time an edge is picked, all neighboring edges
are removed.
The winner is the last player to pick an edge.
6
2
Which player has
a winning strategy
in this graph?
4
1
7
5
3
Similar game: ARC KAYLES
ARC KAYLES: Played on a graph.
●
●
●
Players take turns picking edges from the graph.
Each time an edge is picked, all neighboring edges
are removed.
The winner is the last player to pick an edge.
6
2
4
1
A
7
5
3
Similar game: ARC KAYLES
ARC KAYLES: Played on a graph.
●
●
●
Players take turns picking edges from the graph.
Each time an edge is picked, all neighboring edges
are removed.
The winner is the last player to pick an edge.
6
2
4
1
B
7
5
3
Similar game: ARC KAYLES
ARC KAYLES: Played on a graph.
●
●
●
Players take turns picking edges from the graph.
Each time an edge is picked, all neighboring edges
are removed.
The winner is the last player to pick an edge.
6
2
4
Player A wins!
1
A
7
5
3
Turn-based SET as ARC KAYLES
Turn-based SET is a restriction of ARC KAYLES played
in 3-uniform SET hypergraphs.
Unfortunately:
●
●
Complexity of ARC KAYLES on 3-uniform general
hypergraphs is unknown.
Complexity of ARC KAYLES on graphs is unknown.
Results
Turn-based SET is at least as hard as ARC KAYLES.
Short Turn-based SET (can player A win in at most r rounds?)
is FPT.
→ Short ARC KAYLES is FPT.
Short ARC KAYLES is FPT
●
Edges picked are a maximal matching (of size ≤ r ).
→Graph has |VC| ≤ 2r.
Short ARC KAYLES is FPT
●
Edges picked are a maximal matching (of size ≤ r ).
→Graph has |VC| ≤ 2r.
IS
VC
Short ARC KAYLES is FPT
●
Edges picked are a maximal matching (of size ≤ r ).
→Graph has |VC| ≤ 2r.
●
Twin edges e,e’ : same outcome if e →e’ in a play.
IS
VC
e
Short ARC KAYLES is FPT
●
Edges picked are a maximal matching (of size ≤ r ).
→Graph has |VC| ≤ 2r.
●
Twin edges e,e’ : same outcome if e →e’ in a play.
IS
VC
e’
e
twin vertices
Short ARC KAYLES is FPT
●
Edges picked are a maximal matching (of size ≤ r ).
→Graph has |VC| ≤ 2r.
●
Twin edges e,e’ : same outcome if e →e’ in a play.
IS
●
VC
●
# of twin sets = 22r.
Only their size
matters (if < r ).
Short Turn-based SET
●
Similar trick on 3-uniform hypergraphs?
IS
HS
e
e’
Short Turn-based SET
●
Similar trick on 3-uniform hypergraphs?
IS
HS
N[ ]?
e
e’
Short Turn-based SET
Interesting fact: In SET hypergraphs, a pair of
vertices can belong to at most one hyperedge.
Short Turn-based SET
Interesting fact: In SET hypergraphs, a pair of
vertices can belong to at most one hyperedge.
(For each pair of cards there is a unique third
one which completes the set.)
Short Turn-based SET
●
Similar trick on 3-uniform SET hypergraphs?
IS
HS
N[ ]?
e
e’
SET Summary – Open Problems
SUMMARY:
●
We defined a Turn-based 2 player SET game and
established connections with ARC KAYLES.
OPEN PROBLEMS:
●
●
●
Complexity of Turn-based SET?
Proving the hardness of ARC KAYLES settles the
complexity of the above problem (interesting open
question on its own accord).
Parameterized complexity of ARC KAYLES in general
3-uniform hypergraphs parameterized by r?
The Game of
Joint with Michael Lampis, Kazuhisa Makino, and Yushi Uno
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
●
Each player is dealt a number of cards.
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
●
Player 1
Player 2
Player 3
Drop Pile
UNO - Rules
Deck of cards:
c colors;
● b numbers.
●
Each player is dealt a number of cards.
● They take turns placing cards on the drop pile
following the matching rule: cards agree either in
color or in number.
● The first player unable to play a card loses.
●
Player 1
Player 2
Player 3
Drop Pile
Multiplayer UNO - graph formulation
An UNO instance can be formulated as a p-partite
directed graph:
●
–
Cards↔vertices.
–
Two vertices are joined iff one can be discarded after
the other.
Decision question: Is there a strategy for player 1 to
avoid losing no matter what the other players do?
→Multiplayer UNO is like playing Vertex Generalized
Geography on the UNO (directed p-partite) graph.
(Vertex) Generalized Geography
Rules:
●
●
●
(Di-)Graph G(V,E), starting vertex v.
Players take turns moving a token from vertex to
vertex.
–
Vertex version: Vertices cannot be visited twice;
–
Edge version: Edges cannot be followed twice;
First player unable to make a move loses.
Generalized Geography - Example
Generalized Geography - Example
P1
Generalized Geography - Example
P2
Generalized Geography - Example
P1
Generalized Geography - Example
P2
Generalized Geography - Example
P1
Generalized Geography - Example
Player 1 wins !
Generalized Geography - Complexity
2 Players
●
●
Directed Case: PSPACE-Complete (even on Bipartite
Graphs). [Papadimitriou]
Undirected Case [Fraenkel et al. 1993]
–
Vertex GG: Polynomially solvable.
–
Edge GG: PSPACE-Complete (in P when restricted
to bipartite graphs).
Multiplayer UNO - Complexity
●
2-player UNO is in P (via a reduction to Undirected
Vertex Generalized Geography on bipartite graphs).
[Demaine et al. 2014]
p-player UNO for p ≥ 3 is PSPACE-complete (via a
reduction from Directed Edge Generalized
Geography on bipartite graphs).
Multiplayer UNO - Complexity
●
2-player UNO is in P (via a reduction to Undirected
Vertex Generalized Geography on bipartite graphs).
[Demaine et al. 2014]
p-player UNO for p ≥ 3 is PSPACE-complete (via a
reduction from Directed Edge Generalized
Geography on bipartite graphs).
Directed Edge Geo on bipartite graphs
≤
≤ Directed p-player Edge Geo on p-partite graphs
UNO Summary – Open Problems
SUMMARY:
●
Reformulation of multiplayer UNO as
generalized geography + complexity results.
OPEN PROBLEMS:
●
●
Is short multiplayer UNO FPT?
(Parameterized) complexity of multiplayer UNO
if one attribute is constant / parameter?
Overall Discussion / Questions
●
Games as problems programme is neat! :-)
●
Studying parameterized complexity of games is useful!
Question
●
Find a “meaningful” way to define multiplayer (|p|>2)
combinatorial games (→algorithmic game theory).
Overall Discussion / Questions
●
Games as problems programme is neat! :-)
●
Studying parameterized complexity of games is useful!
Question
●
Find a “meaningful” way to define multiplayer (|p|>2)
combinatorial games (→algorithmic game theory).
THANK
YOU!
I) Michael Lampis and Valia Mitsou: The Computational Complexity of the
Game of Set and its Theoretical Applications. LATIN 2014.
II) Michael Lampis, Kazuhisa Makino, Valia Mitsou, and Yushi Uno:
Parameterized Edge Hamiltonicity. WG 2014 & DAM 2017 (to appear).
III) Valia Mitsou: The Computational Complexity of Some Games and
Puzzles with Theoretical Applications. PhD thesis, 2014.