Abstract. The intersection graph of a set of geometric objects is defined as agraph G = (S; E) in which there is an edge between two nodes si; sj 2 S if si " sj 6 =;. The problem of computing a maximum independent set in the in-tersection graph of a set of objects is known to be NP-complete for most casesin two and higher dimensions. We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R 2. Specifically, given a set of n line segments in the plane with maximum indepen-dent set of size ^, we present algorithms that find an independent set of size atleast ( i) (^=2 log(2n=^)) 1=2 in time O(n

We provide an O(log n)-approximation algorithm for the following problem. Given a convex n-gon P, drawn on a convex piece of paper, cut P out of the piece of paper in the cheapest possible way. No polynomial-time approximation algorithm was known for this problem posed in 1985.

Given a convex n-gon P drawn on a piece of paper Q of unit diameter we prove that it can be cut with a total cost of O(log n). This bound is shown to be asymptotically tight: a regular n-gon (whose circumscribed circle has radius, say, 1/3) drawn on a square piece of paper of unit diameter requires a cut cost of 37 n).