1. Approximate Algorithms (chap. 35)
Motivation:
Many problems are NP-complete, so unlikely find efficient algorithms
Three ways to get around:
If input size is small, exponential algorithm is OK.
Isolate important special case and find poly algorithms for them.
Find near-optimal solutions in poly time.
So approximate algorithms:
An algorithm returning a near-optimal solution is called approximate algorithm.

2. Vertex-cover problem
Vertex cover: given an undirected graph G=(V,E), then a subset V'V such that if (u,v)E, then uV' or v V' (or both).
Size of a vertex cover: the number of vertices in it.
Vertex-cover problem: find a vertex-cover of minimal size.

3. Vertex-cover problem
Vertex-cover problem is NP-complete. (See section ).
Vertex-cover belongs to NP.
Vertex-cover is NP-hard (CLIQUEPvertex-cover.)
Reduce <G,k> where G=<V,E> of a CLIQUE instance to <G',|V|-k> where G'=<V,E'> where E'={(u,v): u,vV, uv and <u,v>E} of a vertex-cover instance.
So find an approximate algorithm.

4. Approximate Ratio
C* is the cost of optimal solution and C is the cost of an approximate algorithm
(n)=max(C/C*, C*/C) where n is size of problem input
If (n)=1, then the algorithm is an optimal algorithm
The larger (n), the worse the algorithm

5. Approximate vertex-cover algorithm

6. Example of approximate vertex-cover

7. 2-approximate vertex-cover
Theorem 35.1 (page 1026).
APPROXIMATE-VERTEX-COVER is a poly time 2-approximate algorithm, i.e., the size of returned vertex cover set is at most twice of the size of optimal vertex-cover.
Proof:
It runs in poly time
The returned C is a vertex-cover.
Let A be the set of edges picked in line 4 and C* be the optimal vertex-cover.
Then C* must include at least one end of each edge in A and no two edges in A are covered by the same vertex in C*, so |C*||A|.
Moreover, |C|=2|A|, so |C|2|C*|.

8. Travelling-saleman problem with triangle inequality

9. Another kind of approximate algorithm
Approximate string matching (also called string matching allowing errors):
Find all the substrings in text T that are close to pattern P.
Edit distance: P is said to be of distance k to a string Q if P can be transformed to Q with k following character operations: insertion, deletion, and substitution.
May have other operations and different operations have different costs.
Refer to the handout paper by Sun Wu and Udi Manber.

10. Algorithms Sequential Parallel Approximate deterministic Random
Probabilistic
Genetic
Evolution and optimization