Ch09 _2 Approximation algorithm 2010/6/17

NP-Complete Problem Enumeration Branch an Bound Greedy Approximation PTAS K-Approximation No Approximation

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.

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 2010/6/17

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. 2010/6/17

(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 9-41 Planar Graphs. 2010/6/17

Terminology The unbounded faces are called exterior faces and all other faces are called interior faces. exterior face interior face 2010/6/17

Terminology We can use faces to mark the level of each node. 1 1 2 2 3 2010/6/17

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 9-43 An Example of 2-Outerplanar Graph. 2010/6/17

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. 2010/6/17

Example A Planar Graph which Has 9 Levels. The Graph Obtained by Removing Nodes in levels 3, 6 and 9. 2010/6/17

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). 2010/6/17

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| (1 - ) = |SOPT| nodes. 2010/6/17

PTAS Therefore, |Sr|  |SOPT| . According to our algorithm, or e =  |SAPX|  |Sr|  |SOPT| or e =  Thus if we set k = -1, then the above formula becomes e  =  E . 2010/6/17

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. 2010/6/17