Efficient approximation algorithms for maximum coverage with group budget constraints

Abstract

Given a ground set U with a non-negative weight wi for each i∈U, a positive integer k and a collection of sets S, which is partitioned into a family of disjoint groups G, the goal of the Maximum Coverage problem with Group budget constraints (MCG) is to select k sets from S, such that the total weight of the union of the k sets is maximized and at most one set is selected from each group G∈G. We first present an approximation algorithm with a factor 1−1e and an exponential time via randomized linear programming rounding technique. Then we improve the time complexity of the algorithm to O((m+n)3.5L+k3.5q7L) for |S|=m, |U|=n, and L being the length of the input, by the key idea of modeling the selection of groups as computing a constrained flow in a corresponding auxiliary graph. The algorithm is later shown can be extended to solve two generalizations of MCG. Last but not the least, we present another algorithm with a time complexity O((m+n)3.5L+kδ10.5L) and a slightly increased approximation ratio 1−e1δ−1 mainly based on the idea of partition, where δ≥2 is a parameter tuning which can balance the time complexity and the ratio.