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

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Pancakes,Puzzles,and PolynomialsCracking the
Cracker Barrel Game

• Christopher Frost
• Michael Peck

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The Cracker Barrel Game
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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?

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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?

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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.

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ComplexityA Look at How Growth Rates Compare
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Complexity ClassesThe 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

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Example NP-Complete Problems

• Protein Folding
• Traveling Salesperson
• Map coloring
• Cracker Barrel?

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Project Goal

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

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Proving NP-Completeness

• Must show two conditions

• Problem belongs to NP
• Is at least as hard as any problem in NP

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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.

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Proving NP-CompletenessSolving 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
Answer
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3-SAT to CB Transformer

• Represent a logical expression on a peg board.
• (x1 ? x2 ? x4) ? (?x1 ? x2 ? x3)

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3-SAT to CB Transformer Inside The Mysterious
Blue Tile
Blue Tile Goal Allow green peg across iff yellow
has come down.
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3-SAT to CB Transformer
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3-SAT to CB Transformer Inside The Grey Tile
x
1
Grey Tile Goal Allow both the green and yellow
peg across at any time.
x
1
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3-SAT to CB Transformer
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3-SAT to CB Transformer Which Term?
Term-Area Goal Allow peg for a pair of terms to
represent either the variable or its negation as
true.
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3-SAT to CB Transformer
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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
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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, PNP
• If not, P?NP

(Given that CB is NP-complete)
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Questions?