Finding near optimal separators in planar graphs

A k-ratio edge separator is a set of edges which separates a weighted graph into two disconnected sets of components neither of which contains more than k-1/k of the original graph´s weight. An optimal quotient separator is an edge separator where the size of the separator (i.e., the number of edges) divided by the weight of the smaller set of components is minimized. An optimal quotient k-ratio separator is an edge separator where the size of the separator (i.e., the number of edges) divided by the smaller of either 1/k of the total weight or the weight of the smaller set of components is minimized. In this paper we present an algorithm that finds the optimal quotient k-ratio separator for any k ≥ 3. We use the optimal quotient algorithm to obtain approximation algorithms for finding optimal k-ratio edge separators for any k ≥ 3. Given a planar graph with a size OPT k-ratio separator, we describe an algorithm which a finds k-ratio separator which costs less than O(OPT log n). More importantly the algorithm finds ck-ratio separators (for any c ≫ 1) which cost less than C(c)OPT, where C(c) depends only on c.