Leveraging Optimal Transport to Design Optimal Mechanisms for the Facility Location Problem

The facility location problem is a classical combinatorial optimization problem with extensive applications spanning communication technology, economic management, traffic governance, and public services. The facility location problem is to assign a set of clients to a set of facilities such that each client connects to a facility and the total cost (open cost and connection cost) is as low as possible. Among its various models, the uncapacitated facility location (UFL) problem is the most fundamental and widely studied. However, in real-world scenarios, resource constraints often make the UFL problem insufficient, necessitating more generalized models. This investigation primarily focuses on the universal facility location (Uni-FL) problem, a generalized framework encompassing both capacitated facility location problems (with hard and soft capacity constraints) and the UFL problem. Through a systematic analysis, we examine the Uni-FL problem alongside its specialized variants: the hard capacitated facility location (HCFL) problem and soft capacitated facility location (SCFL) problem. A comprehensive survey is conducted of existing approximation algorithms and theoretical results. The relevant results of their important variants are also discussed. In addition, we propose some open questions and future research directions for this problem based on existing research.

Clustering is a ubiquitous problem that arises in many applications in different fields such as data mining, image processing, machine learning, and bioinformatics. Clustering problems have been extensively studied as optimization problems with various objective functions in the Operations Research and Computer Science literature. We focus on a class of objective functions more commonly referred to as facility location problems. These problems arise in a wide range of applications such as, plant or warehouse location problems, cache placement problems, and network design problems where the costs obey economies of scale. In the simplest of these problems, the uncapacitated facility location (UFL) problem, we want to open facilities at some subset of a given set of locations and assign each client in a given set D to an open facility so as to minimize the sum of the facility opening costs and client assignment costs. This a very well-studied problem; however it fails to address many of the requirements of real applications. In this thesis we consider various problems that build upon UFL and capture additional issues that arise in applications such as, uncertainties in the data, clients with different service needs, and facilities with interconnectivity requirements. By focusing initially on facility location problems in these new models, we develop new algorithmic techniques that will find application in a wide range of settings. We consider a widely used paradigm in stochastic programming to model settings where the underlying data, for example, the locations or demands of the clients, may be uncertain: the 2-stage with recourse model that involves making some initial decisions, observing additional information, and then augmenting the initial decisions, if necessary, by taking recourse actions. We present a randomized polynomial time algorithm that solves a large class of 2-stage stochastic linear programs (LPs) to near-optimality with high probability. We exploit this tool to devise the first approximation algorithms for various 2-stage discrete stochastic problems such as the stochastic versions of the set cover, vertex cover, and facility location problems, when the underlying random data is only given as a “black box” and no restrictions are placed on the cost structure. We introduce the facility location with service installation costs problem to model applications involving clients with different service requirements. if the service requested by it has been installed at the facility (incurring a service installation cost). The connected facility location problem captures settings where the open facilities want to communicate with each other or with a central authority; we model this by requiring that the open facilities be interconnected by a Steiner tree. We give intuitive and efficient algorithms for both these problems. We use these algorithms to obtain approximation algorithms for the κ-median variants of these problems, where in addition to all of the constraints of the problem, a bound of κ is imposed on the number of facilities that may be opened.

A new approximation algorithm for the [formula omitted]-facility location problem

This thesis addresses the development of geometric approximation algorithms for huge datasets and is subdivided into two parts. The first part deals with algorithms for facility location problems, and the second part is concerned with the problem of computing compact representations of finite metric spaces. Facility location problems belong to the most studied problems in combinatorial optimization and operations research. In the facility location variants considered in this thesis, the input consists of a set of points where each point is a client as well as a potential location for a facility. Each client has to be served by a facility. However, connecting a client incurs connection costs, and opening or maintaining a facility causes so-called opening costs. The goal is to open a subset of the input points as facilities such that the total cost of the system is minimized. We are particularly interested in facility location problems for large-scale distributed systems of mobile objects. In order to be able to analyze such complex systems, we examine the following partial aspects: • At first, we present a distributed algorithm that, in case of uniform opening costs for the facilities and uniform demands of the clients, computes in only three communication rounds a constant-factor approximation for the metric facility location problem. • In Chapter 4, we introduce a mobile facility location problem where the input points move continuously in a constant-dimensional Euclidean space. In contrast to Chapter 3, we also take non-uniform opening costs for the facilities and non-uniform demands of the clients into account. We propose an event-driven data structure that efficiently maintains a subset of the mobile points as open facilities such that, at any time, the total cost of the system is at most a constant factor larger than the optimal facility location cost. • In Chapter 5, we consider again a uniform facility location problem. However, this time, we develop a streaming algorithm where the input stream consists of insert and delete operations of points from a constant-dimensional Euclidean space. While reading the input stream, our algorithm maintains a summary of the current point set in a subtle way with the result that the required space is polylogarithmic in the size of the input stream and, at any time, it can output a constant-factor approximation of the optimal facility location cost. • In the next chapter, we give an efficient streaming implementation of a k-means clustering algorithm. The k-means clustering problem is closely related to the facility

Approximation algorithms for facility location problems with a special class of subadditive cost functions

In this paper we devise the stochastic and robust approaches to study the soft-capacitated facility location problem with uncertainty. We first present a new stochastic soft-capacitated model called The 2-Stage Soft Capacitated Facility Location Problem and solve it via an approximation algorithm by reducing it to linear-cost version of 2-stage facility location problem and dynamic facility location problem. We then present a novel robust model of soft-capacitated facility location, The Robust Soft Capacitated Facility Location Problem. To solve it, we improve the approximation algorithm proposed by Byrka et al. (LP-rounding algorithms for facility-location problems. CoRR, 2010a) for RFTFL and then treat it similarly as in the stochastic case. The improvement results in an approximation factor of $$\alpha + 4$$ for the robust fault-tolerant facility location problem, which is best so far.

Approximation algorithm for squared metric facility location problem with nonuniform capacities

We consider the nth power metric facility location problem with linear penalties (M $$^n$$ FLPLP) in this work, extending both the nth power metric facility location problem (M $$^n$$ FLP) and the metric facility location problem with linear penalties (MFLPLP). We present an LP-rounding based approximation algorithm to the M $$^n$$ FLPLP with bi-factor approximation ratio $$(\gamma _f, \gamma _c)$$ , where $$\gamma _f$$ and $$\gamma _c$$ are the ratios corresponding to facility, and connection and penalty costs respectively. Finally we show that the bi-factor curve is close to the lower bound $$(\gamma _f, 1 + (3^n - 1) e^{-\gamma _f})$$ when the facility factor $$\gamma _f > 2$$ for the M $$^2$$ FLPLP.

We investigate the power of randomness in the context of a fundamental Bayesian optimal mechanism design problem - a single seller aims to maximize expected revenue by allocating multiple kinds of resources to agents with preferences drawn from a known distribution. When the agents' preferences are single-dimensional Myerson's seminal work [14] shows that randomness offers no benefit - the optimal mechanism is always deterministic. In the multi-dimensional case, where each agent's preferences are given by different values for each of the available services, Briest et al.[6] recently showed that the gap between the expected revenue obtained by an optimal randomized mechanism and an optimal deterministic mechanism can be unbounded even when a single agent is offered only 4 services. However, this large gap is attained through unnatural instances where values of the agent for different services are correlated in a specific way. We show that when the agent's values involve no correlation or a specific kind of positive correlation, the benefit of randomness is only a small constant factor (4 and 8 respectively). Our model of positively correlated values (that we call the common base value model) is a natural model for unit-demand agents and items that are substitutes. Our results extend to multiple agent settings as well.

The power of randomness in Bayesian optimal mechanism design

Given m facilities each with an opening cost, n demands, and distance between every demand and facility, the Facility Location problem flnds a solution which opens some facilities to connect every demand to an opened facility such that the total cost of the solution is minimized. The k-Facility Location problem further requires that the number of opened facilities is at most k, where k is a parameter given in the instance of the problem. We consider the Facility Location problems satisfying that for every demand the ratio of the longest distance to facilities and the shortest distance to facilities is at most !, where ! is a predeflned constant. Using the local search approach with scaling technique and error control technique, for any arbitrarily small constant † > 0, we give a polynomial-time approximation algorithm for the !-constrained Facility Location problem with approximation ratio 1 + p ! + 1 +†, which signiflcantly improves the previous best known ratio (! + 1)=fi for some 1 6 fi 6 2, and a polynomial-time approximation algorithm for the !-constrained k- Facility Location problem with approximation ratio !+1+†. On the aspect of approximation hardness, we prove that unless NP µ DTIME(n O(log log n) ), the !-constrained Facility Location problem cannot be approximated within 1 + ln p ! i 1, which slightly improves the previous best known hardness result 1:243 + 0:316ln(! i 1). The experimental results on the standard test instances of Facility Location problem show that our algorithm also has good performance in practice.

In this paper, we introduce and study the Non-Uniform k-Center (NUkC) problem. Given a finite metric space (X, d) and a collection of balls of radii {r_1 >= ... >= r_k}, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation alpha, such that the union of balls of radius alpha*r_i around the i-th center covers all the points in X. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds, or as a wireless router placement problem with routers of different powers/ranges. The NUkC problem generalizes the classic k-center problem when all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k-center with outliers (kCwO for short) problem when there are k balls of radius 1 and l balls of radius 0. There are 2-approximation and 3-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no O(1)-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an (O(1), O(1))-bi-criteria approximation result: we give an O(1)-approximation to the optimal dilation, however, we may open Theta(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community. We show NUkC is as hard as the firefighter problem. While we don't know if the converse is true, we are able to adapt ideas from recent works [Chalermsook/Chuzhoy, SODA 2010; Asjiashvili/Baggio/Zenklusen, arXiv 2016] in non-trivial ways to obtain our constant factor bi-criteria approximation.

In the \begin{document}$ k $\end{document} -facility location problem, an important combinatorial optimization problem combining the classical facility location and \begin{document}$ k $\end{document} -median problems, we are given the locations of some facilities and clients, and need to open at most \begin{document}$ k $\end{document} facilities and connect all clients to opened facilities, minimizing the facility opening and connection cost. This paper considers the squared metric \begin{document}$ k $\end{document} -facility location problem with linear penalties, a robust version of the \begin{document}$ k $\end{document} -facility location problem. In this problem, we do not have to connect all clients to facilities, but each client that is not served by any facility must pay a penalty cost. The connection costs of facilities and clients are supposed to be squared metric, which is more general than the metric case. We provide a constant approximation algorithm based on the local search scheme with add, drop, and swap operations for this problem. Furthermore, we improve the approximation ratio by using the scaling technique.

New algorithms for fair k-center problem with outliers and capacity constraints

The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the smoothed approximation ratio to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the average-case approximation ratio to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, Filos-Ratsikas et al. [2014] show that, amongst all truthful mechanisms, random priority achieves the tight approximation ratio bound of Theta(sqrt{n}). We prove that, despite of this worst-case bound, random priority has a constant smoothed approximation ratio. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to 1+e. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is Theta(sqrt{n}) times of what random priority achieves. These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis.