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Approximate Algorithms (chap. 35)PowerPoint Presentation

Approximate Algorithms (chap. 35)

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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.

Vertex-cover problem

- Vertex cover: given an undirected graph G=(V,E), then a subset V'V such that if (u,v)E, then uV' 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.

Vertex-cover problem

- Vertex-cover problem is NP-complete. (See section 34.5.2).
- Vertex-cover belongs to NP.
- Vertex-cover is NP-hard (CLIQUEPvertex-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,vV, uv and <u,v>E} of a vertex-cover instance.

- So find an approximate algorithm.

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

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*|.

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.

Algorithms

- Sequential
- Parallel
- Approximate
- deterministic
- Random
- Probabilistic
- Genetic
- Evolution and optimization

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