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Exam 2 Review

Reductions

We say that a problem A reduces to a problem B if there is a polynomial time reduction function f such that for all x, x \in A \iff f(x) \in B.

To prove a reduction, we need to show that the reduction function f:

1. runs in polynomial time
2. x \in A \iff f(x) \in B.

Useful results from reductions

1. B is at least as hard as A if A \leq B.
2. If we can solve B in polynomial time, then we can solve A in polynomial time.
3. If we want to solve problem A, and we already know an efficient algorithm for B, then we can use the reduction A \leq B to solve A efficiently.
4. If we want to show that B is NP-hard, we can do this by showing that A \leq B for some known NP-hard problem A.

P is the class of problems that can be solved in polynomial time. NP is the class of problems that can be verified in polynomial time.

We know that P \subseteq NP.

NP-complete problems

A problem is NP-complete if it is in NP and it is also NP-hard.

NP

A problem is in NP if

1. there is a polynomial size certificate for the problem, and
2. there is a polynomial time verifier for the problem that takes the certificate and checks whether it is a valid solution.

NP-hard

A problem is NP-hard if every instance of NP hard problem can be reduced to it in polynomial time.

List of known NP-hard problems:

1. 3-SAT (or SAT)
2. Independent Set
3. Vertex Cover
4. 3-coloring
5. Hamiltonian Cycle
6. Hamiltonian Path

Approximation Algorithms

Randomized Algorithms

Online Algorithms