NP Complete Introduction

Objectives

Notes

• This is very informal, it is not rigorous
• We begin to talk about problems, not algorithms.
• P is the set of problems that we can solve in polynomial time.
  • There exists an algorithm with performance O(n^k), k an integer constant, the size of the input.
• NP is a set of problems that we can check in polynomial time
  • Ie if C is a solution (or certificate) to a problem.
  • There is an algorithm A(I, C) , O(n^k) for constant integer k, that determines if C is the correct solution for input I.
• An example
  • The input to a sorting problem is the array A
  • The output of a sorting algorithm is A', the ordered array
  • What algorithm can we use to check to see if A' is in order?
  • Is this a polynomial time algorithm?
  • The Sorting is an element of the set NP
• Any problem in P is also in NP

  Check(Input, Certificate)
  Use the polynomial time algorithm to find the solution to the given input
  Check to see if this matches the certificate
     

• Therefore P ⊆ NP
• The subset NP-Complete (NPC) is a set of problems that
  • Are in NP
  • And are as hard as any other problem in NP
• In some sense here we are not looking at how easy a problem is, but how hard it is.
  • Think the lower bounds on sorting.
• We frequently look at decision problems
  • These are problems which given a set if input produce a yes or a no.
  • Given a circuit, can a set of input be found to make the circuit true?
  • Given a graph, can I find a cycle that visits every vertex exactly once?
• These problems are easy to test,
  • I take the circuit and the input that supposedly produces a true and evaluate it.
  • I trace the proposed cycle in the graph and see if it visits all vertexes.
  • Decision problems can be mapped to optimization problems
    • We transform an optimization into a series of "Can this be done in time 1, in time 2, ... up to time n, the size of the input.
    • Note this is a polynomial loop, O(n) + time to solve problem.
  • We will be somewhat sloppy here too, decision, optimization, whatever
  • Reductions
    • Let A and B both be decision problems.
    • A problem A can be reduced to a problem B, A ≤ B
    • There is an algorithm that transforms the data for A to data for B
    • The output from B (yes, no) is also the output for B.
    • If this can be done in polynomial time, A ≤_pB
  • If we can reduce one problem to another, it is no harder than the second
    • To find the k^th item in data we could

      Sort the list of data
      return data[k]
            

    • Thus the task of finding the k^th item is no harder than sorting.
    • (I know this is not a decision problem, but it will do for now)
  • So if I can reduce any problem, in polynomial time, to a polynomial problem, it is also polynomial.
    • Finding the k^th element ≤_P sorting
    • Sorting ⊂ P
    • Thus Finding the k^th element ⊂ P
  • Generally we
    • Can show a problem is in P by producing an algorithm.
    • show something is in NP by showing we can check it in P and not having an algorithm that shows it is in P.
  • In a graph, can we find a path that visits every vertex of a graph exactly once? This is the Hamiltonian path problem or HAM-PATH
    • We don't know a good way to solve it, so it is not in P
    • We do know a good way to check it, so it is in NP
  • NP-Complete is a way to try to fix this.
  • Ie if we can reduce any problem in the set NP-COMPLETE to a problem P in polynomial time,
  • Then we solve P in polynomial time
  • We can solve every problem in NP in polynomial time.
  • We actually have a large set of problems in NP-Complete
    • Wikipedia has a list.
    • But this is not a full list.
    • We will look at some of these problems on Monday
  • But we don't know how to solve ANY of these problems quickly.
  • A few more notes
    • NP are hard problmes we can check easily
    • NP-Complete are hard problems we can check easily and all problems in NP can be reduced to them
    • NP-Hard are problems that eveything in NP-Complete can be reduced to, but we might not be able to check them easily
    • 
  • So the question is: does P = NP?