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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.

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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 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.

### Vertex-cover problem

• Vertex-cover problem is NP-complete. (See section 34.5.2).

• 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.

### 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