NOVEL GENERALIZATIONS AND ALGORITHMS FOR THE MAX-k-COVERAGE PROBLEM

Author

Luc Cote, Dartmouth College

Date of Award

Spring 2023

Document Type

Thesis (Undergraduate)

Department

Computer Science

First Advisor

Deeparnab Chakrabarty

Abstract

In this thesis we consider the fundamental optimization problem known as the Max-k- Coverage problem and its generalizations. We first discuss the well-studied generalization to the problem of maximizing a monotone submodular function subject to a cardinality constraint and introduce a new primal-dual algorithm which achieves the optimal approximation factor of (1 − 1/e). While greedy algorithms have been known to achieve this approximation factor, our algorithms also provide a dual certificate which upper bounds the optimum value of an instance. This certificate may be used in practice to provide much stronger guarantees than the worst-case (1 − 1/e) approximation factor. We then introduce a novel generalization of the Max-k-Coverage problem known as the Monotone Coverage Utility Maximization (mcum) problem. Here, we are given a set of clients C and a set of facilities F in a metric space (F ∪ C, d), a positive integer k, and a radius parameter r. Each client v ∈ C is associated with a non-decreasing utility function f_v : N → R_≥0, and the objective is to locate centers S ⊆ F with |S| = k so as to maximize ∑_v∈C f_v(S ∩ B_v(r)), where B_v(r) is the ball of radius r around v. This general problem reduces to the special case Threshold Utility Maximization (thrum), where the f_v’s are threshold step functions, that is, f_v(a) = w_v if a ≥ ℓ_v for some positive integer ℓ_v, and 0 otherwise. thrum, in turn, generalizes Max-k-Coverage problem by putting ℓ_v = 1 for all clients. On the other hand, even when all ℓ_v’s are 2, the problem is as hard as Densest-k- Subgraph. We circumvent this hardness by addressing the “soft” version of the problem, where the same center can be opened multiple times. In this realm, we give a 1 2 -approximation in the uniform case where all ℓ_v’s equal ℓ, which readily implies a Θ(1/ log k) approximation for the general problem. We also describe matching integrality gaps for the natural LP relaxations for both problems. Furthermore, we consider the thrum problem from a radius-dilation perspective. Here, we want to obtain a utility of opt, but allow the algorithm’s neighborhood to be dilated to αr for α ≥ 1, with α as small as possible. This problem becomes equivalent to the Fault- tolerant k-Supplier with Outliers (FkSO) problem. Our main result is a 3-approximation for FkSO in the uniform case where all ℓ_v = ℓ, which improves upon an 11-approximation described by Inamdar and Varadarajan in 2020.

Recommended Citation

Cote, Luc, "NOVEL GENERALIZATIONS AND ALGORITHMS FOR THE MAX-k-COVERAGE PROBLEM" (2023). Computer Science Senior Theses. 24.
https://digitalcommons.dartmouth.edu/cs_senior_theses/24