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The Computational Complexityof Games and Puzzles
Valia Mitsou City University
of New York
Games and Puzzles …
… with Theoretical Applications
● Generalization of wolf-goat- cabbage puzzle is similar to Vertex Cover.
● Generalizations of SET game are similar to well known graph-theoretic problems.
● Generalization of UNO game is similar to Hamiltonicity.
Computational Complexity
We assume a background in Complexity Theory:➢ P vs NP➢ Beyond NP (PSPACE)➢ Algorithms vs Hardness➢ Approximation
Parameterized Complexity Primer
➢Parameterized Complexity: Except from input n, we consider an additional variable k which is called the parameter.
➢Parameterized Complexity differentiates between algorithms that run in time nf(k) and those that run in time g(k)·nc.
➢The 1st class is called XP and the 2nd is called FPT.
Parameterized Complexity Primer
➢FPT algorithms are usually much faster than those that run in time nf(k).
➢W-hard: Problems in XP considered not to admit FPT algorithms.
➢There is a reduction program similar to the theory of NP-Completeness for proving W-hardness of parameterized problems.
Outline
➢ The wolf-goat-cabbage puzzle➢ The game of SET➢ The game of UNO➢ The game of Scrabble
The Wolf-Goat-Cabbage puzzle
Joint work with Michael Lampis
≤ man + 1
How can they all cross the river safely?
“Previous Work”
• “Propositiones ad acuendos iuvenes”, Alcuin of York, 8th century A.D.
• We propose a generalization of Alcuin’s puzzle.
Our generalization
Our generalization
• We seek to transport n items, given their incompatibility graph.
• Objective: Minimize boat size.• We call this the Ferry Cover Problem
OPTFC (G) ≥ OPTVC (G)
OPTFC (G) ≥ OPTVC (G)
OPTFC (G) ≥ OPTVC (G)
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
OPTFC (G) ≤ OPTVC (G) + 1
The Ferry Cover Problem
LM 2007:
OPTVC (G) ≤ OPTFC (G) ≤ OPTVC (G) + 1
Hardness and Approximation
LM 2007:• Ferry Cover is NP-hard and APX-hard (like
Vertex Cover).• A ρ-approximation algorithm for Vertex
Cover yields a (ρ + 1/OPTFC)-approximation algorithm for Ferry Cover.
Small and Large Boat Graphs
Graphs are divided into two categories:• Small-boat, if OPTFC (G) = OPTVC (G)
• Large-boat, if OPTFC (G) = OPTVC (G) + 1
Small and Large Boat Graphs
LM 2007: Stars with >2 leaves are big-boat, all other trees are small-boat.
CHW 2008: Categorization of Bipartite, Chordal, and Planar graphs.
CHW 2008: Deciding if a general graph is small-boat is NP-hard.
Trip Constrained Ferry Cover
• Variation of Ferry Cover: we are also given a trip constraint.
• We seek to minimize the size of the boat s.t. there is a solution within this constraint.
FC1 (3 trips max allowed)
LM 2007:
• FC1 is APX-hard.
• 2-approximation (trivial).• 4/3-approximation for bipartite graphs.• 1.56-approximation for planar graphs.
Further Work
• Can we have an efficient non-trivial approximation of FC1 in the general case?
• Are there any graph families for which Vertex Cover is easy but Ferry Cover is hard?
• Is it in NP to determine whether a graph is small boat?
The Game of
Joint work with Michael Lampis
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
There are 34 = 81 different cards in total (one for each combination of values).
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 number;
✔ all have different shadings.
The Game of Set - Rules
● Deal 12 cards;
● Find a valid set.
Naive way to find a set
● Search among all possible triples.
Generalization: k-value 1SET
● Input: m cards, n attributes, k values
● Question: Does there exist a valid set of k cards with all values the same or all values different?
● In the original game m=12, n=4 and k=3.
Hypergraph formulation
Cards = Hyper-edges
Attributes = Dimensions# of Values = size of Parts
n a t t r i b u t e s
In SET, hyperedges are allowed to overlap as long as they all overlap on the same value.
k
values
Connection with n-Dim. Matchingn d i m e n s i o n s
Perfect n-Dimensional Matching:Given a hypergraph H(V,E), pick k hyperedgessuch that all vertices are covered exactly once.
k
size
Complexity Results for k-value 1SET
● For k unbounded:
– n = 2 → P1 (find a star or a bipartite matching)
– n ≥ 3 → NP-Complete1
1. Chaudhuri et al: On the complexity of the game of set (manuscript 2003).
Complexity Results for k-value 1SET
● For n unbounded:
– k parameter → XP
k-value 1SET parameterized by k is W-hard
perfect n-DM parameterized by k is W-hard
(n-DM with unbounded number of possible values parameterized by both n and k is FPT)2
2. Fellows et al: Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems (Algorithmica 2008).
Reduction
From k-Multicolored Clique
● Input: k-partite graph, each part of size n
● Question: Does there exist a clique of size k?
● Parameter: k
Reduction
From k-Multicolored Clique
● Input: k-partite graph, each part of size n
● Question: Does there exist a clique of size k?
● Parameter: k
k-Multicolored Clique is W-hard
Construction
The constructed multigraph:
● n·k(k-1) dimensions, in groups of size n
● k+ possible different values
k
21 2 3
Construction
The constructed multigraph:
● For each vertex we construct a v-hyperedge.
Example shows construction for 3rd vertex from part 1.
Construction
The constructed multigraph:
● For each edge we construct an e-hyperedge.
Example shows edge connecting 1st vertex of part i with 2nd vertex of part j.
Correctness
● For each edge eij between parts i and j, the 3 hyperedges corresponding to i, j and ij cover the respective values entirely.
Example
Corollary for k-1SET
● Constructed hyperedges cannot all overlap, unless they correspond to the same parts.
● If there exists a SET, it is also a perfect matching (and vice versa).
k-1SET parameterized by k is W-hard
Multi-round variations
● Naive algorithm works for complete enumeration of all co-existing valid sets -without card removal.
Multi-round variations
● However, finding the max number of disjoint valid sets -with card removal- is not easy (even for k=3).
Multi-round variations
● However, finding the max number of disjoint valid sets -with card removal- is not easy (even for k=3).
Multi-round variations
● Opposite goal: find the min number of valid sets that destroys all existing sets.
Multi-round variations as hypergraph problems
Construct a 3-uniform hypergraph: vertices ↔ cards, hyperedges ↔ valid sets.
Multi-round variations as hypergraph problems
Construct a 3-uniform hypergraph: vertices ↔ cards, hyperedges ↔ valid sets.
● max 3-value rSET: restriction of 3-Set Packing to SET hypergraphs).
● min 3-value rSET: restriction of Edge Dominating Set to SET hypergraphs).
Both restrictions remain NP-hard.min-3-rSET parameterized by r is FPTAlgorithm works for Edge Dominating Set on general 3-uniform hypergraphs.
A 2-player 3-value rSET Game
RULES:
● Deal m 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;
A 2-player 3-value rSET Game
RULES:
● Deal m 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 1 Player 2
A 2-player 3-value rSET Game
RULES:
● Deal m 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 1 Player 2
A 2-player 3-value rSET Game
RULES:
● Deal m 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 1 Player 2
Player 1 wins!
A similar 2 player game
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.
1
2
35
62
4
5
6
7
Which player has a winning strategy in this graph?
A similar 2 player game
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.
1
2
35
62
4
5
6
7P1
A similar 2 player game
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.
1
2
35
62
4
5
6
7P2
A similar 2 player game
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.
1
2
35
62
4
5
6
7P1Player 1 wins!
2 player SET as ARC-KAYLES
2-player 3-value rSET 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.
Facts
● Similar 2 player game NODE-KAYLES is PSPACE-C3 and AW[*]-hard when parameterized by the # of rounds r4.
● Complexity of ARC-KAYLES on general hypergraph with hyperedges of unbounded size is PSPACE-C5.
3. Schaefer: On the complexity of some two-person perfect-information games (JCSS 1978).
4. Abrahamson et al: Fixed parameter tractability and completeness IV: On completeness for W[P] and PSPACE analogues (Ann. Pure Appl. Logic 1995).
5. Daniel Grier: Deciding the Winner of an Arbitrary Finite Poset Game Is PSPACE-Complete. (ICALP (1) 2013).
Results
2p-3-rSET is at least as hard as ARC-KAYLES.
2p-3-rSET with parameter r (can player 1 win in at most r rounds?) is FPT.
Algorithm works on more general 3-uniform h-graphs, where each pair of vertices has unique 3rd that they all belong in the same hyperedge.
Conclusions – Open Problems
CONCLUSIONS:● We studied the complexity of one- and multi- round variations of
game of SET.● We established connections of the game with Multi-Dimensional
Matching, Set Packing, Edge Dominating Set, and Arc Kayles.
OPEN PROBLEMS:● Complexity of 2p 3-rSET?● 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 work with Michael Lampis, Kaz Makino, and Yushi Uno
UNO - Rules
Deck of cards each with 2 attributes (colors, numbers) + special cards; p ≥ 2 players;
● Players are dealt 7 cards. ● They take turns discarding cards.● Next discarded card should match the previous discarded card either in color or in number.
● If a player doesn't have a matching card, she can draw one card from a draw pile.
● First player to get rid of all her cards wins.
Combinatorial Model
Unbounded - Perfect information - No chance
● Card attributes: c colors, b numbers (no special cards);
● p ≥ 1 players;● Players are dealt mi cards each;● Everyone's cards are open to everyone;● If a player doesn't have a matching card, she loses the
game (no draw pile);
Combinatorial Model
Goal:
● Solitaire version: Get rid of all cards in hand.
● Multi-player version: First player who can't discard a card loses.
Graph formulation
An UNO instance can be formulated as a p-partite graph:– Cards are vertices;
– Two vertices are joined iff one can be discarded after the other;
Solitaire version ↔ Hamiltonian Path
Multiplayer version ↔ Generalized Geography
Solitaire UNO as Hamiltonicity
Solitaire UNO as Hamiltonicity
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO as Edge-Hamiltonicity
g
b
r
y
2
3
4
5
Colors Numbers
Edge-Hamiltonian Path:
“Ordering of the edges such that consecutive edges share an common attribute.”
Solitaire UNO – Results
● Bertossi 1981: The edge-Hamiltonian path problem is NP-complete => Solitaire UNO is NP-complete.
● DDHUUU 2014: Solitaire UNO with 2 attributes is in XP (parameterized by the number of colors c).
● LMMU 2014: Solitaire UNO with r attributes is FPT (parameterized by the number of colors c); when r = 2, it even admits a cubic kernel.
Kernelization algorithms
Given an instance of the problem, we construct (in polynomial time) an equivalent new instance with size that depends only on the parameter.
Solitaire UNO with 2 attributes admits a cubic kernel
Given an instance of UNO with c colors and b numbers, we construct (in polynomial time) an equivalent new instance where:– For each color, there will be at most 4c2 numbers of
that color.
– Thus, there will be at most 4c3 numbers.
Solitaire UNO with 2 attributes admits a cubic kernel
From an EHP SOL, we can construct an EHP SOL' where each color-group appears at most c times.
SOL: 1 5 5 3 … 2 8 8 7
Solitaire UNO with 2 attributes admits a cubic kernel
From an EHP SOL, we can construct an EHP SOL' where each color-group appears at most c times.
SOL':1 5 8 2 … 3 5 8 7
Solitaire UNO with 2 attributes admits a cubic kernel
From an EHP SOL, we can construct an EHP SOL' where each color-group appears at most c times.
SOL': … … …
At most c
At most c-1
Solitaire UNO with 2 attributes admits a cubic kernel
Goal (reminder): Construct an equivalent instance where for each color, there will be at most 4c2 numbers of that color.
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers> 4c2
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers> 4c2
S
Sm
allove
rlap
N(S)g ≤ 4c
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers> 4c2
S
Sm
allove
rlap
N(S)
7
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers> 4c2
S
Sm
allove
rlap
N(S)
7
colors
Graph that contains 7 has an EHP SOL
Graph that doesn't contain 7 has an EHP SOL'.
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers≥ 4c2
S
Sm
allove
rlap
N(S)
7
SOL':
At most c
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers≥ 4c2
S
Sm
allove
rlap
N(S)
7
SOL': 7
At most c
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers> 4c2
S
Sm
allove
rlap
N(S)
7
SOL: … 7 …
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
7
SOL: … 7 7 7 …
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
SOL: … 7 7 7 …
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
7
SOL: … 4 7 9 …
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
SOL: … 4 7 9 …
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
7
SOL: … 4 7 7 …
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
> 4c7
SOL: … 4 7 7 …
y
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
> 4c7
SOL: … 4 7 7 …
y
At most c
At most c
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
> 4c7
SOL: … 4 7 7 …
y
At most 2c
At most 2c
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
72
SOL: … 4 7 7 … 2 2
y
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Blue numbers
S
Sm
allove
rlap
N(S)
7
SOL: … 4 2 7 2 7 …
y
colors
Solitaire UNO with 2 attributes admits a cubic kernel
bBlue numbersS
Sm
allove
rlap
N(S)
Ø
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Sm
allove
rlap
… Blue
numbers≤ 4c2t
pg
colors
Solitaire UNO with 2 attributes admits a cubic kernel
b
Sm
allove
rlap
…
t
pg
→ At most 4c3 numbers in total.
colors
Blue numbers≤ 4c2
Multiplayer UNO - graph formulation
An UNO instance can be formulated as a p-partite graph:– Cards are vertices;
– Two vertices are joined iff one can be discarded after the other.
Multiplayer UNO is like playing Generalized Geography on the UNO graph.
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;
● Last player to make a move wins.
New York → Kyoto → Osaka → Athens → Sapporo
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 - Results
● Directed Case: PSPACE-Complete (even on Bipartite or Planar Graphs).
● Undirected Case– Vertex GG: Polynomially solvable;
– Edge GG: PSPACE-Compelte (in P when restricted to bipartite graphs).
Multiplayer UNO
● DDHUUU 2014: 2-player UNO is in P (by a reduction to Undirected Vertex Generalized Geography in bipartite graphs).
● M 2014: p-player UNO for p≥3 is PSPACE-complete (by a reduction from Directed Edge Generalized Geography in bipartite graphs).
Multiplayer UNO is PSPACE-complete
Proof Sketch:
Directed Edge Geo on bipartite graphs ≤ Directed p-player Edge Geo on p-partite graphs ≤ p-player UNO
Multiplayer UNO is PSPACE-complete
Directed p-player Edge Geo on p-partite graphs:
● p players play on their respective part (i → i+1);
● Can player 1 avoid losing no matter what the other players do?
Multiplayer UNO is PSPACE-complete
Proof Sketch:
Directed Geo on bipartite graphs ≤ Directed p-player Geo on p-partite graphs
Multiplayer UNO is PSPACE-complete
Proof Sketch:
Directed p-player GEO on p-partite graphs ≤ p-player UNO
● For each directed edge (x,y) belonging to player i in GEO we give (in UNO):
– player i the cards (x, xy) and (y, xy);
– to all other players j the card (j, xy). ● If v V∈ 1 is the starting node in GEO, set the starting
UNO card to be (v, e) for some edge e V∈ 1 × V2.
Multiplayer UNO is PSPACE-complete
Proof Sketch:
Directed p-player GEO on p-partite graphs ≤ p-player UNO
● Since v V∈ 1 and e V∈ 1 × V2, then player 1 should pick a card (v, vu), u V∈ 2 to start the UNO game.
● This move corresponds to picking edge (v,u) in GEO.
Multiplayer UNO is PSPACE-complete
Proof Sketch:
Directed p-player GEO on p-partite graphs ≤ p-player UNO
● When player i plays edge (x,y) in GEO, this will force a whole round in UNO that will finish again in player i:
– No other player (but players i-1, i) can have a card with its first attribute being x.
– Player i+1 will have to play card (i+1, xy) …
Multiplayer UNO is PSPACE-complete
Proof Sketch:
Directed p-player GEO on p-partite graphs ≤ p-player UNO
● When player i plays edge (x,y) in GEO, this will force a whole round in UNO that will finish again in player i.
● When the round is complete, player i+1 is free to choose the next card (that corresponds to picking the next edge in GEO).
Summary
We continued the work from DDHUUU 20014:● We showed that solitaire UNO parameterized
by the number of colors is FPT.– This is true even for unbounded attributes.
● We showed that multiplayer UNO for 3 or more players is PSPACE-complete.
Future Work
● For the multiplayer version:– Is the problem FPT if size of one attribute is a
parameter?
– Is the problem FPT if we consider a short version (up to k rounds with k being a parameter)?
– Define a more realistic model of UNO (the winner in the original game might be the loser in the current model).
The game of
Joint work with Michael Lampis and Karolina Soltys
A famous game...
● Word game played on a grid● Over 150 million sets sold in 121 countries
in 29 languages.
Scrabble Components - Rules
● 2-4 players● Board● Bag● Letters● Rack● Turn
Difficulty to make a turn - Intuition
Two main task:
Difficulty to make a turn - Intuition
Two main task:● Form a word
Difficulty to make a turn - Intuition
Two main task:● Form a word● Place it on the board
Difficulty to make a turn - Intuition
Two main task:● Form a word● Place it on the board
Difficulty to make a turn - Intuition
Two main task:● Form a word● Place it on the board
Both tasks contribute independently to the hardness of the problem.
Our Model
● Generalized version (unbounded board)● Deterministic, full-information● Made-up language● No special squares on the board● All tiles worth the same amount of points
Our Results
● Deterministic generalized version of Scrabble is PSPACE-Complete
● Hardness comes from two different aspects of the game:
– Formation
– Placement
● Game is hard even for constant-size alphabet, word, rack.
● 1-player version is NP-Complete
Future Work
● Find the minimum alphabet size for which the problem becomes hard;
● Prove that Scrabble is PSPACE-Complete for the English dictionary;
● Study the randomized version of the game.
Conclusions
● We studied four games and puzzles and showed hardness and algorithmic results.
● We established that three of them have interesting connections with well-know combinatorial problems.
● Our results progressed the state of the art for some of these problems (Multi-dimensional Matching, Edge-Dominating Set, Edge-hamiltonian Path).
Future QuestionsConcrete problems:
● Study the complexity of different games (complexity of Connect 4?)
● Study other games connected to problems (complexity of Flow?)
Future QuestionsMore general questions:
● Define multiplayer models in a more realistic way so that players can be selfish.
● Allow chance and imperfect information.