the computational complexity of games and puzzles · the computational complexity of games and...

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) + 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


Each attribute can take one of 3 values:

– Symbol● Oval● Diamond● Squiggle



– Color● Red● Green● Purple



– Shading● Blank● Stripped● Solid



– Number● One● Two● Three


There are 34 = 81 different cards in total (one for each combination of values).


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.


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.


● 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


● For each vertex we construct a v-hyperedge.

Example shows construction for 3rd vertex from part 1.

Construction


● 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.


● However, finding the max number of disjoint valid sets -with card removal- is not easy (even for k=3).


● 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;


RULES:

● Deal m cards;




Player 1 Player 2


RULES:

● Deal m cards;




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?





1

2

35

62

4

5

6

7P1





1

2

35

62

4

5

6

7P2





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 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 – 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.


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



SOL':1 5 8 2 … 3 5 8 7



SOL': … … …

At most c

At most c-1


Goal (reminder): Construct an equivalent instance where for each color, there will be at most 4c2 numbers of that color.


b

Blue numbers> 4c2

colors


b

Blue numbers> 4c2

S

Sm

allove

rlap

N(S)g ≤ 4c

colors


b

Blue numbers> 4c2

S

Sm

allove

rlap

N(S)

7

colors


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'.


b

Blue numbers≥ 4c2

S

Sm

allove

rlap

N(S)

7

SOL':

At most c

colors


b

Blue numbers≥ 4c2

S

Sm

allove

rlap

N(S)

7

SOL': 7

At most c

colors


b

Blue numbers> 4c2

S

Sm

allove

rlap

N(S)

7

SOL: … 7 …

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

7

SOL: … 7 7 7 …

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

SOL: … 7 7 7 …

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

7

SOL: … 4 7 9 …

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

SOL: … 4 7 9 …

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

7

SOL: … 4 7 7 …

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

> 4c7

SOL: … 4 7 7 …

y

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

> 4c7

SOL: … 4 7 7 …

y

At most c

At most c

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

> 4c7

SOL: … 4 7 7 …

y

At most 2c

At most 2c

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

72

SOL: … 4 7 7 … 2 2

y

colors


b

Blue numbers

S

Sm

allove

rlap

N(S)

7

SOL: … 4 2 7 2 7 …

y

colors


bBlue numbersS

Sm

allove

rlap

N(S)

Ø

colors


b

Sm

allove

rlap

… Blue

numbers≤ 4c2t

pg

colors


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


P1


P2


P1


P2


P1


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


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?


Proof Sketch:

Directed Geo on bipartite graphs ≤ Directed p-player Geo on p-partite graphs


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.


Proof Sketch:


● 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.


Proof Sketch:


● 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) …


Proof Sketch:


● 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:


Two main task:● Form a word


Two main task:● Form a word● Place it on the board


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.

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