An Approximation Algorithm for the Maximum Independent Set Problem on Planar Graphs

In this paper we consider the maximum independent set problem in which one would like to find a maximum set of independent (i.e., pairwise nonadjacent) vertices in a given graph. The problem is NP-complete, and still remains so even if we restrict ourselves to the class of planar graphs. It has been conjectured that there exist no polynomial-time exact algorithms for any NP-complete problems. We present a polynomial-time approximation algorithm for the maximum independent set problem on planar graphs. For a given planar graph having any number n of vertices, our algorithm finds, in $O(n\log n)$ time, an independent set that is necessarily larger in size than half a maximum independent set. Thus the absolute worst case ratio of our algorithm is greater than $\tfrac{1}{2}$.

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