Improved PTASs for Geometric Covering and Domination Problems with Cardinality Constraints

We studied the following two geometric optimization problems with cardinality constraints. • Capacitated covering with unit balls: Given a set of n points in R and a positive integer L, partition them into a minimum number of groups such that each group contains at most L points, and can be covered by a unit ball in R. • Capacitated connected dominating set in unit disk graph: Given a positive integer L and a unit disk graph G = (V,E), find a multi-set P of vertices with minimum size such that P is connected, and can dominate all vertices in V if each vertex in P can only be used to dominate L vertices. For the first problem, we provide a (1 + ε)-approximation algorithm with n d−1) running time when d is a fixed constant. 1 This improves the state-of-the-art PTAS with running time n ) by Ghasemi and Razzazi [GR14]. For the second problem, we provide a (1 + ε)approximation with n running time. The running times for both algorithms essentially match the state-of-the-art PTASs for the uncapacitated versions: the n d−1) time algorithm for covering with unit balls problem by Feder and Greene [FG88] and Gonzalez [Gon91]; and the n time algorithm for minimum dominating set in unit disk graph by Nieberg and Hurink [NH05]. Our results build on a structure property introduced by Ghasemi and Razzazi [GR14], and a recursive partition scheme, resembling the renowned PTAS for Euclidean TSP by Arora [Aro96].