Pancakes, Puzzles, and Polynomials: Cracking the Cracker Barrel Game

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Pancakes, Puzzles, and Polynomials: Cracking the Cracker Barrel Game. Christopher Frost Michael Peck. The Cracker Barrel Game. The Cracker Barrel Problem (CB).

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### Pancakes,Puzzles,and Polynomials:Cracking the Cracker Barrel Game

Christopher Frost

Michael Peck

The Cracker Barrel Problem (CB)

Given an arbitrarily sized board with some initial configuration of pegs, is there a sequence of jumps such that the board will be left with one remaining peg?

How Hard Is It To Solve The Cracker Barrel Game?
• Straightforward way of solving the peg board puzzle:
• Try all possible ways to move a peg
• Look at all possible ways of moving a peg for each of the above moves
• ...
• Until find a sequence of moves with one peg left or run out of possible moves (no solution)
• How long will this take to solve?
• Is this the fastest way?
Complexity
• Measuring complexity:
• How does the time needed to solve a problem grow as the size of the input to the problem grows?
• Example: linear-time
• If the size of the input doubles, the time needed to solve doubles.

P

NP

NP-C

or

NP-C

P=NP

• Does P=NP? Are all the problems in NP also in P?
• The biggest unanswered question in computer science.
Complexity Classes:The Big Three
• Problems that can be solved in nk time
• Problems that can be verified in nk time
• Problems that are at least as hard as all other problems in NP
• P – Polynomial
• NP – Nondeterministic Polynomial
• NP-Complete
Example NP-Complete Problems
• Protein Folding
• Traveling Salesperson
• Map coloring
• Cracker Barrel?
Project Goal

Is CB (the Cracker Barrel problem) NP-Complete?

Proving NP-Completeness
• Must show two conditions:
• Problem belongs to NP
• Is at least as hard as any problem in NP
Example NP-Complete Problem: 3-SAT

Expression

Clauses

Terms

• (x1 x2 x4)  (x1 x2 x3)
• Is there an assignment of values to these terms that makes the above expression true?
• Yes!
• One solution: If x1 = true and x3 = true, the above expression is true.
Proving NP-Completeness:Solving any problem in NP using CB
• Reduction: Showing that a known NP-complete problem can be solved using a solver for CB.

3-SAT Solver

CB Solver

3-SAT to CB Transformer

Input to 3-SAT Solver

Ø

Ø

Ø

Ø

x

x

x

x

x

x

x

x

1

2

3

4

1

2

3

4

x

1

x

C

2

1

x

4

Ø

x

1

x

C

2

2

x

3

3-SAT to CB Transformer
• Represent a logical expression on a peg board.
• (x1 x2 x4)  (x1 x2 x3)

The Non-transitive

Peg Hierarchy

of Power

=

>

1.

2.

>

>

a > b: a can jump b,

but b can’t jump a.

x

x

1

1

3-SAT to CB Transformer: Inside The Mysterious Blue Tile

Blue Tile Goal:

Allow green peg across

iff yellow has come down.

The Non-transitive

Peg Hierarchy

of Power

=

>

1.

2.

>

>

a > b: a can jump b,

but b can’t jump a.

3-SAT to CB Transformer: Inside The Green Tile

Green Tile Goal:

Reduce the number of green pegs to one

iff every clause had one or more pegs cross the board.

x

1

C

x

2

1

x

4

Progress and Implications
• Progress:
• Our best known CB solver takes exponential time
• Proved a variation of CB is NP-Complete
• Implications:
• Is it possible to create a CB solver that runs in polynomial time?
• If so, P=NP
• If not, P≠NP

(Given that CB is NP-complete)