NP-Completeness

• We want to talk about how "bad" an algorithm can become
• All of the algoithms we have studied so far are polynomial time algorithms
• O(n^k) where k is an integer that does not depend on the input.
• k above is constant
• These problems are considered tractable
• Problems for which no polynomial time algorithm exists are considered to be intractable
• The set of all problems that can be solved with a polynomial time algorithm is called the set P
• A decision problem is a problem where yes/no answer is appropriate
• All problems can be mapped (or turned into) decision problems.
  • Consider an electronic cirucit.
  • It has n boolean inputs, and one boolean output.
  • Find a set of inputs which will produce a True for an output?
  • One way to do this is try all possible permutations of the input.
  • This is O(n!) if there are n inputs.
  • This is not in class P.
  • This is known as Circuit-Satisfiability.
• A verification algorithm is an algorithm that verifies an answer is correct given the answer and a key.
  • For example, given a circit and a set of inputs, we can determine if the circuit evaluates to true.
  • We can even do this in polynomial time.
  • Just try the input on the circuit.
• In the above example, the input to the circuit is a certificate
• If we can verify an answer, given a certificate, in polynomial time the algorithm belongs to a class of problems called NP
• So P is the set of algorithms that can be performed in polynomial time.
• NP is the set of algorithms who's output can be verified in polynominal time.
• It seems that these are two distinct sets of algorithms.
• It is not known if P=NP, this is a major open question in CS
• But there are some big prizes if you can show this.