WK15. Vertex Cover and Approximation Algorithm - PowerPoint PPT Presentation

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WK15. Vertex Cover and Approximation Algorithm

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1. choose a edge A, kick out all edges which connect to 2 ends of A ( include A). 2. choose other edges and repeat step1 until all edges are kick out. ... – PowerPoint PPT presentation

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Title: WK15. Vertex Cover and Approximation Algorithm


1
WK15. Vertex Cover and Approximation Algorithm
  • By Lin, Jr-Shiun
  • Choi, Jae Sung

2
Vertex Cover
  • Definition
  • A set of vertices in an undirected graph
    where every edge connects at least one vertex.
    The vertex cover problem is to find a minimum
    size set and is NP-complete. (NIST)

3
Vertex Cover
2
  • Example
  • Determine the smallest subset of vertex that
    Cover the graph on the right.

1
3
4
5
4
Vertex Cover
2
  • Example
  • Determine the smallest subset of vertex that
    Cover the graph on the right.
  • ANS 1, 3, 4

1
3
4
5
5
Approximation Algorithm
  • Minimization problem
  • if F , C gt 1 such that you can find a
    solution which is lt CA in polynomial time.
  • then this is a approximation solution
  • Maximization problem
  • if F , C gtlt 1 such that you can find a solution
    which is gt CA in polynomial time.
  • then this is a approximation solution

6
Approximation Algorithm for Vertex Cover
  • Algorithm
  • 1. choose a edge A, kick out all edges which
    connect to 2 ends of A ( include A).
  • 2. choose other edges and repeat step1 until
    all edges are kick out.
  • vertex-cover that is atmost twice the size of
    an optimal cover (rmuhamma)

7
Euclidean TSP
  • Example

TempP
Approx T
MST
8
Euclidean TSP
  • Let Optimal TSP T
  • Let MST M
  • (TSPspanning tree that visits all vertex)
  • M should be smaller than T
  • T gtM
  • TempP2m
  • Approx T lt2Mlt2T
  • Approx T lt (1 X )T
  • X is any small number, e.g. 0.000001

9
Approximation Algorithm for TSP
  • TSP is general graph that can not be
    approximated.
  • Claim an algorithm can solve it.
  • we will show that approximation algorithm can be
    used to solve the Hamiltonian cycle problem

G
Tlt CT
10
Approximation Algorithm for TSP
CN1
Add weight
G
Approximation Algorithm
CN1 lt CT 5N1lt5T (N of Vertex) 26lt5T
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