Shifting strategy for geometric graphs without geometry

Abstract

We give a simple framework which is an alternative to the celebrated and widely used shifting strategy of Hochbaum and Maass (J. ACM 32(1):103---136, 1985) which has yielded efficient algorithms with good approximation bounds for numerous optimization problems in low-dimensional Euclidean space. Our framework does not require the input graph/metric to have a geometric realization--it only requires that the input graph satisfy some weak property referred to as growth boundedness. Growth bounded graphs form an important graph class that has been used to model wireless networks. We show how to apply the framework to obtain a polynomial time approximation scheme (PTAS) for the maximum (weighted) independent set problem on this important graph class; the problem is W[1]-complete. Via a more sophisticated application of our framework, we show how to obtain a PTAS for the maximum (weighted) independent set for intersection graphs of (low-dimensional) fat objects that are expressed without geometry. Erlebach et al. (SIAM J. Comput. 34(6):1302---1323, 2005) and Chan (J. Algorithms 46(2):178---189, 2003) independently gave a PTAS for maximum weighted independent set problem for intersection graphs of fat geometric objects, say ball graphs, which required a geometric representation of the input. Our result gives a positive answer to a question of Erlebach et al. (SIAM J. Comput. 34(6):1302---1323, 2005) who asked if a PTAS for this problem can be obtained without access to a geometric representation.