Pancakes, Puzzles, and Polynomials: Cracking the Cracker Barrel Game Christopher Frost Michael Peck

2 The Cracker Barrel Game

3 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?

4 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?

5 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.

6 Complexity: A Look at How Growth Rates Compare

7 P – Polynomial NP – Nondeterministic Polynomial NP-Complete Complexity Classes: The Big Three P NP P=NP or Does P=NP? Are all the problems in NP also in P? The biggest unanswered question in computer science. NP-C Problems that can be solved in n k time Problems that can be verified in n k time Problems that are at least as hard as all other problems in NP

8 Example NP-Complete Problems Protein Folding Traveling Salesperson Map coloring Cracker Barrel?

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

10 Must show two conditions: Problem belongs to NP Is at least as hard as any problem in NP Proving NP-Completeness

11 Example NP-Complete Problem: 3-SAT (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. Expression ClausesTerms

12 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. CB Solver 3-SAT Solver 3-SAT to CB Transformer Answer Input to 3-SAT Solver

13 3-SAT to CB Transformer Represent a logical expression on a peg board. (x1  x2  x4)  (  x1  x2  x3) C 1 C 2 x 1 x 1 x 2 x 4  x 1 x 2 x 3  x 1 x 2 x 3 x 4  x 2  x 3  x 4

14 3-SAT to CB Transformer: Inside The Mysterious Blue Tile x 1 x 1 Blue Tile Goal: Allow green peg across iff yellow has come down. > The Non-transitive Peg Hierarchy of Power a > b: a can jump b, but b can’t jump a = > >

15 3-SAT to CB Transformer

16 > The Non-transitive Peg Hierarchy of Power a > b: a can jump b, but b can’t jump a = > > 3-SAT to CB Transformer: Inside The Grey Tile x 1 x 1 Grey Tile Goal: Allow both the green and yellow peg across at any time.

17 3-SAT to CB Transformer

18 > The Non-transitive Peg Hierarchy of Power a > b: a can jump b, but b can’t jump a = > > 3-SAT to CB Transformer: Which Term? x 1 Term-Area Goal: Allow peg for a pair of terms to represent either the variable or its negation as true. x 1 

19 3-SAT to CB Transformer

20 > The Non-transitive Peg Hierarchy of Power 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. C 1 x 1 x 2 x 4

21 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? (Given that CB is NP-complete) If so, P=NP If not, P≠NP

22 Questions?