Lecture 03-27-17

Sawmill problem

• Cut into specific pieces
• Costs calculated by size of board
• Example:

  • 4

    3

    7

    2

    8

  • ABCD: 24 + 20 + 17 + 10
  • BACD: 24 + 7 + 17 + 10
  • CDBA: 24 + 10 + 14 + 7
• Minimum total cost
• Very similar to matrix multiplication & optimal binary tree

Office Hours 3/28

• 9:00am - 10:30am, 1:00pm - 2:30pm

Proving a problem is difficult

• P: decision problems that can be solved in polynomial time
• NP: can very an answer of yes in polynomial time

P = NP?

• What makes it important?
  1. Many important problems are known to be in NP but not known to be in P
  2. Many important problems are known to be in NP and are in P iff P = NP

Evidence of difficulty

1. Problem is in NP iff P = NP
2. We believe P != NP
3. Therefore, problem cannot be solved in polynomial time
   • Wrong proof
   • Hard problem: NP-complete
   • Complete: hardest problem in its class

Reduction:

• Problem A => Problem B
• Any instance of A into an instance of B
• Show A is at least as hard as B
  • A => B or B => A?
  • Answer: B => A
• Answer to instance of y must be same as answer to instance of x
• X => Y
• Reduction must take polynomial time (and space)
• X infinity_p Y

NP-Complete

• X is NP-complete if
  1. X is in NP
  2. Every problem in NP polynomially reduced to X
• To show:
  1. Take known NP-complete and turn into X
• Chicken & egg problem