A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomial-time approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for N P-hard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersection graphs of other "disk-like" geometric objects (such as squares or regular polygons), also in higher dimensions. Key words. independent set, vertex cover, shifting strategy, disk graph AMS subject classifications. 68Q25, 68R10 DOI. 10.1137/S0097539702402676 1. Introduction. Intersection graphs are graphs whose vertices are represented by sets such that two vertices are adjacent if and only if the corresponding sets have a nonempty intersection. They have been studied by many authors [12, 6, 24] . We are interested in approximation algorithms for N P-hard optimization problems on intersection graphs of geometric objects, in particular approximation algorithms for independent set and vertex cover. Two prominent applications of geometric intersection graphs are frequency assignment in cellular networks [13, 22] and map labeling [1] . The goal of the maximum weight independent set problem (MWIS) is to compute, for a given set of geometric objects with certain weights, a subset of disjoint (nonoverlapping) objects with maximum total weight. The goal of the minimum weight vertex cover problem (MWVC) is to compute a subset of the given objects with minimum total weight such that, for any two intersecting objects, at least one of the objects is contained in the subset. MIS and MVC refer to the unweighted versions of these problems. We obtain polynomial-time approximation schemes for MWIS and MWVC in the intersection graphs of disks, squares, or other "disk-like" objects, also in higher dimensions.
Combined with a dynamic programming approach, the shifting strategy also gives a PTAS for the Maximum Independent Set problem on Disk Graphs with arbitrary radii REF .
Polynomial-Time Approximation Schemes for Geometric Intersection Graphs
SIAM J. Comput.
SIAM J. Comput.
Mathematics Computer Science