NP-Completeness

• This is from CLRS
• Most of the algorithms we have studied are polynomial time O(n^k) where k is constant.
• Can all problems be solved in polynomial time?
  • Clearly computing all permutations can not
  • Some can't be done at all.
    • The halting problem: Given a description of an arbitrary computer program, decide whether the program finishes running or continues to run forever.
    • A creative proof
  • So are there other classes of problems?
  • Polynomial algorithms
    • Class P
    • as expressed above can be solved in O(n^k) for any set of input.
  • We will examine a class (or set) of problems called NP.
    • Given a solution, the solution can be checked in polynomial time.
  • Does P=NP?
    • Unknown
    • No one has been able to show this since the early 70s.
• Slight changes to problems may mean major changes in the complexity of finding the solution.
  • The 0-1 knapsack problems asks what mix of items should be placed in a knapsack, of limited capacity, to maximize value.
    • Each item has a different value and weight.
    • This is in class NP
    • Algorithm: Brute force, try everything.
  • The fractional knapsack problem asks what mix of items should be placed in a knapsack, of limited capacity, to maximize value.
    • Items may have different values.
    • But all items are of the same size.
    • This is in class P.
    • Algorithm: Greedy take the most valuable items first.
  • Given a directed graph G=(V,E).
    • A Euler tour traverses each edge exactly once.
      • Does an Euler tour exist?
      • Fleury's Algorithm, (1883) solves this in O(|E|²), but better algorithms exist.
      • So this problem is in class P.
    • A Hamiltonian cycle visits each vertex (other than the first) exactly once.
      • Determining the existence of a Hamiltonian cycle is in class NP.
• The difference between problems in class P and problems in class NP is verification.
  • You can solve the Eulerian Path question for a given graph G in O(n^k)
  • You can not, as far as we know, solve the Hamiltonian Path question for a given graph G in O(n^k)
  • However, if you are given a path H, you can test to see if it is a Hamiltonian path in O(n^k).
  • Or you can verify a solution in polynomial time.
  • Any problem in P is in NP.
  • But is P⊆NP or is P⊂NP?
• NP-Complete
  • This is the set of hardest problems in the class NP.
  • A problem is in this set if EVERY OTHER PROBLEM in the set NP can be reduced to the problem in polynomial time.
  • In other words:
    • B is in NP-Complete if
    • For every other problem A in the set NP
    • Transform A to B in polynomial time.
    • Solve B
    • Transform the answer back from B to A in polynomial time.
  • A solution to B provides a solution to A.
  • If you can show that B is in P, then EVERYTHING in NP is in P, and P = NP
  • On the other hand, if you can show that B is not in P, then EVERYTHING else in NP-Complete can not be in P, thus P ≠ NP.
  • Read quote on 968
• Decision Problems
  • Many hard problems are optimization problems.
  • Like the knapsack problem.
  • Much work in NP-Complete is with decision problems.
  • Does a Hamiltonian path exist?
  • Optimization problems can be cast as decision problems
  • Can a solution be found with value k? (IE for 0-1 knapsack, can a value of k be loaded into the knapsack from the items?)
  • Decision problems are easier than optimization problems.
  • But if the decision problem is hard, the optimization problem is hard. (Otherwise, we would just optimize the question).
  • And if the decision problem is easy, the optimization problem is probably easy as well.
  • So the work we will do is in decision problems, not optimization problems.
• Reductions
  • Or showing that a given problem is no more difficult than another problem.
  • Consider only decision problems.