## 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