Shifting strategy for geometric graphs without geometry

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.