Generalized Budgeted Submodular Set Function Maximization
In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements.
We first present an algorithm that guarantees an approximation factor of 1/2(1-1/e^alpha), where alpha <= 1 is the approximation factor of an algorithm for a sub-problem. We give two polynomial-time algorithms to solve this sub-problem. The first one gives us alpha=1- epsilon if the costs satisfies a specific condition, which is fulfilled in several relevant cases, including the unitary costs case and the problem of maximizing a monotone submodular function under a knapsack constraint. The second one guarantees alpha=1-1/e-epsilon for the general case. The gap between our approximation guarantees and the known inapproximability bounds is 1/2.
We extend our algorithm to a bi-criterion approximation algorithm in which we are allowed to spend an extra budget up to a factor beta >= 1 to guarantee a 1/2(1-1/e^(alpha beta))-approximation. If we set beta=1/(alpha)ln (1/(2 epsilon)), the algorithm achieves an approximation factor of 1/2-epsilon, for any arbitrarily small epsilon>0.
Submodular set function
Approximation algorithms
Budgeted Maximum Coverage
Theory of computation~Approximation algorithms analysis
Theory of computation~Packing and covering problems
31:1-31:14
Regular Paper
http://arxiv.org/abs/1808.03085
Francesco
Cellinese
Francesco Cellinese
Gran Sasso Science Institute, L'Aquila, Italy
Gianlorenzo
D'Angelo
Gianlorenzo D'Angelo
Gran Sasso Science Institute, L'Aquila, Italy
Gianpiero
Monaco
Gianpiero Monaco
University of L’Aquila, L'Aquila, Italy
Yllka
Velaj
Yllka Velaj
University of Chieti-Pescara, Pescara, Italy
10.4230/LIPIcs.MFCS.2018.31
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Francesco Cellinese, Gianlorenzo D'Angelo, Gianpiero Monaco, and Yllka Velaj
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