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Approximate Algorithms (chap. 35)

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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  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 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.

  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. 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.

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

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