1. Ch09 _2 Approximation algorithm
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2. NP-Complete Problem Enumeration Branch an Bound Greedy Approximation
PTAS
K-Approximation
No Approximation

3. Polynomial-Time Approximation Schemes
A problem L has a polynomial-time approximation
scheme (PTAS) if it has a polynomial-time
(1+ε)-approximation algorithm, for any fixed ε >0 (this value can appear in the running time).
For example, there is a PTAS for finding the maximum independent set problem on planar graphs.

4. Independent set
An independent set is a set of vertices in a graph, no two of which are adjacent.
An maximal independent set is an independent set that is not a subset of any other independent set.
maximum independent sets
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5. Finding the maximum independent set problem
The input is an undirected graph, and the output is a maximum independent set in the graph.
It is a NP-hard problem and it is also hard to approximate, and the decision problem is NP- Complete.
Fortunately, there is a PTAS for finding the maximum independent set problem on planar graphs.
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6. (a) A Planar Graph. (b) A Graph Which Is Not Planar.
A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. 1 5 2 4 3
(a) A Planar Graph (b) A Graph Which Is Not Planar.
Figure Planar Graphs.
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7. Terminology
The unbounded faces are called exterior faces and all other faces are called interior faces.
exterior face
interior face
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8. Terminology We can use faces to mark the level of each node. 1 1 2 2 3
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9. Figure 9-43 An Example of 2-Outerplanar Graph.
Terminology
A graph is k-outerplanar if it has no nodes with level greater than k.
Figure An Example of 2-Outerplanar Graph.
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10. Property
Given an arbitrary planar graph G, we can decompose it into a set of k-outerplanar graphs.
For a k-outerplanar graph, an optimal solution for the maximum independent set problem can be found in O(8kn) time through the dynamic programming approach where n is the number of vertices.
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11. Example A Planar Graph which Has 9 Levels.
The Graph Obtained by Removing Nodes in levels 3, 6 and 9.
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12. An Approximation Algorithm
Step 1. For all i = 0, 1, ... , k, do
(1.1) Let Gi be the graph obtained by deleting all nodes with levels congruent to i (mod k + 1). The remaining subgraphs are all k- outerplanar graphs.
(1.2) For each k-outerplanar graph, find its maximum independent set. Let Si denote the union of these solutions.
Step 2. Among S0 , S1 , ... , Sk , choose the Sj with the maximum size and let it be our approximation solution SAPX .
The time-complexity of our approximation algorithm is obviously O(8kkn).
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13. PTAS
Thus there is at least one r, such that at most of vertices in SOPT are at a level which is congruent to r (mod k + 1).
This means that the solution Sr obtained by deleting the nodes in class r from SOPT will have at least
|SOPT| ( ) = |SOPT| nodes.
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14. PTAS Therefore, |Sr|  |SOPT| . According to our algorithm, or e = 
|SAPX|  |Sr|  |SOPT| or
e = 
Thus if we set k = , then the above formula becomes
e  =  E .
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15. Conclusion
This shows that for every given error bound E, we have a corresponding k to guarantee that the approximation solution differs from the optimum one within this error ratio.
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