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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
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 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 problem1
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
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
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
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
Algorithms
  • Sequential
  • Parallel
  • Approximate
  • deterministic
  • Random
  • Probabilistic
  • Genetic
    • Evolution and optimization
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