NP-Complete Problems

NP and P What is NP? NP is the set of all decision problems (question with yes-or-no answer) for which the 'yes'-answers can be verified in polynomial time (O(n^k) where n is the problem size, and k is a constant) by a deterministic Turing machine. Polynomial time is sometimes used as the definition of fast or quickly. What is P? P is the set of all decision problems which can be solved in polynomial time by a deterministic Turing machine. Since it can solve in polynomial time, it can also be verified in polynomial time. Therefore P is a subset of NP.

Class of “P” Problems Class P consists of (decision) problems that are solvable in polynomial time Polynomial-time algorithms Worst-case running time is O(nk), for some constant k Examples of polynomial time: O(n2), O(n3), O(1), O(n lg n) Examples of non-polynomial time: O(2n), O(nn), O(n!)

Class of “NP” Problems Class NP consists of problems that could be solved by NP algorithms i.e., verifiable in polynomial time If we were given a “certificate” of a solution, we could verify that the certificate is correct in time polynomial to the size of the input Warning: NP does not mean “non-polynomial”

NP-Complete What is NP-Complete? A problem x that is in NP is also in NP-Complete if and only if every other problem in NP can be quickly (ie. in polynomial time) transformed into x. In other words: x is in NP, and Every problem in NP is reducible to x So what makes NP-Complete so interesting is that if any one of the NP-Complete problems was to be solved quickly then all NP problems can be solved quickly

NP-Hard What is NP-Hard? NP-Hard are problems that are at least as hard as the hardest problems in NP. Note that NP-Complete problems are also NP-hard. However not all NP-hard problems are NP (or even a decision problem), despite having 'NP' as a prefix. That is the NP in NP-hard does not mean 'non-deterministic polynomial time'. Yes this is confusing but its usage is entrenched and unlikely to change.

Tractable/Intractable Problems Problems in P are also called tractable Problems not in P are intractable or unsolvable Can be solved in reasonable time only for small inputs Or, can not be solved at all Are non-polynomial algorithms always worst than polynomial algorithms? - n1,000,000 is technically tractable, but really impossible - nlog log log n is technically intractable, but easy

Reductions Reduction is a way of saying that one problem is “easier” than another. We say that problem A is easier than problem B, (i.e., we write “A  B”) if we can solve A using the algorithm that solves B. Idea: transform the inputs of A to inputs of B f Problem B   yes no Problem A

Polynomial Reductions Given two problems A, B, we say that A is polynomially reducible to B (A p B) if: There exists a function f that converts the input of A to inputs of B in polynomial time A(i) = YES  B(f(i)) = YES

NP-Complete Problems

Traveling Salesman

5-Clique

Hamiltonian Path

Map Coloring

Vertex Cover (VC) Given a graph and an integer k, is there a collection of k vertices such that each edge is connected to one of the vertices in the collection?