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.
• The Traveling Salesman Problem (TSP), Given a complete weighted Graph G=(V,E) find the minimal cost simple cycle that visits all nodes.
• The Traveling Salesman Decision Problem(TSP), Given a complete weighted Graph G=(V,E), is there a simple cycle of cost K that visits all nodes?
• One way to do this is to compute all possible cycles.
• This is O(|V|!)
• Definately not in class P
• A verification algorithm is an algorithm that verifies an answer is correct given the answer and a key.
• For example, given a graph, the answer (Yes) and a simple cycle, we can check to see if the cost of the cycle is equal to K.
• In the above example, the simple cycle 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
• Again, we can check the sum of all the paths in the certificate in O(|V|), and we can check to see if all of the verticies are on the path in O(|V|), therefore the TSP is in the class NP
• It is not know if P=NP
• But there are some big prizes if you can show this.