NP-complete Problem 2Prof. S M Lee

Department of Computer Science

Encodings

An encoding "e" is a mapping from a set S to binary strings

Use encodings to map abstract problems to concrete problems

Example - Shortest Path010100101010010101101100010101001010

Hamiltonian cycle of an undirected graph G= (V, E) is a simple cycle that contains each vertex in V

A hamiltonian graph is a graph that has a hamiltonian cycle.

Let m = |V|, it takes m! operations to determine if G= (V, E) is a hamiltonian graph

If Mary claims a graph is hamiltonian graph and provides the vertices in order on the hamiltonian cycle then we can verify her claim in polynomial time

The potential cycle is called the certificate.

Why Are We Studying NP-complete Problems?

Because we want to know what problems can be solved by computers

Polynomial ReductionsNP-complete problems are known as decision problems. This means that a specific item of input data is accepted and depending on the specific problem, it is required to determine if the instance does have the property, then the answer yes is returned; if not, then the answer no is returned

Polynomial ReductionReduction of a problem P to a problem Q: problem q’s answer for t(x) must be the same as p’s answer for x

T Algorithm for Q

x(an input for p)

T(x)

An input for Q

Yes or No answer

A Simple ReductionLet the problem P be: given a sequence of boolean values, does at least one of them have the value trueLet Q be: given a sequence of integers, is the maximum of the integer positive?Let the transformation T be defined by: t(x1,x2,…,xn)=(y1,y2,…,yn) where yi=1 if xi=true, and yi=0 if xi=falseClearly an algorithm to solve Q, when applied to y1,y2,…,yn, solves P for x1,x2,…,xn