Multi-level Facility Location Problems

We conduct a comprehensive review on multi-level facility location problems which extend several classical facility location problems and can be regarded as a subclass within the well-established field of hierarchical facility location. We first present the main characteristics of these problems and discuss some similarities and differences with related areas. Based on the types of decisions involved in the optimization process, we identify three different categories of multi-level facility location problems. We present overviews of formulations, algorithms and applications, and we trace the historical development of the field

Multi-level facility location problems

We study of a class of discrete facility location problems, called multi-level facility location problems, that has received major attention in the last decade. These problems arise in several applications such as in production-distribution systems, telecommunication networks, freight transportation, and health care, among others. Moreover, they generalize well-known facility location problems which have been shown to lie at the heart of operations research due to their applicability and mathematical structure. We first present a comprehensive review of multi-level facility location problems where we formally define and categorize them based on the types of decisions involved. We also point out some gaps in the literature and present overviews of related applications, models and algorithms. We then concentrate our efforts on the development of solution methods for a general multi-level uncapacitated facility location problem. In particular, based on an alternative combinatorial representation of the problem whose objective function satisfies the submodularity property, we propose a mixed integer linear programming formulation. Using that same representation, we present approximation algorithms with constant performance guarantees for the problem and analyze some special cases where these worst-case bounds are sharper. Finally, we develop an exact algorithm based on Benders decomposition for a slightly more general problem where the activation of links between level of facilities is also considered part of the decision process. Extensive computational experiments are presented to assess the performance of the various models and algorithms studied. We show that the multi-level extension of some fundamental problems in operations research maintain certain structure that allows us to develop more efficient algorithms in practice

Approximation algorithms for clustering and facility location problems

In this thesis we design and analyze algorithms for various facility location and clustering problems. The problems we study are NP-Hard and therefore, assuming P is not equal NP, there do not exist polynomial time algorithms to solve them optimally. One approach to cope with the intractability of these problems is to design approximation algorithms which run in polynomial-time and output a near-optimal solution for all instances of the problem. However these algorithms do not always work well in practice. Often heuristics with no explicit approximation guarantee perform quite well. To bridge this gap between theory and practice, and to design algorithms that are tuned for instances arising in practice, there is an increasing emphasis on beyond worst-case analysis. In this thesis we consider both these approaches. In the first part we design worst case approximation algorithms for Uniform Submodular Facility Location (USFL), and Capacitated k-center (CapKCenter) problems. USFL is a generalization of the well-known Uncapacitated Facility Location problem. In USFL the cost of opening a facility is a submodular function of the clients assigned to it (the function is identical for all facilities). We show that a natural greedy algorithm (which gives constant factor approximation for Uncapacitated Facility Location and other facility location problems) has a lower bound of log(n), where n is the number of clients. We present an O(log^2 k) approximation algorithm where k is the number of facilities. The algorithm is based on rounding a convex relaxation. We further consider several special cases of the problem and give improved approximation bounds for them. The CapKCenter problem is an extension of the well-known k-center problem: each facility has a maximum capacity on the number of clients that can be assigned to it. We obtain a 9-approximation for this problem via a linear programming (LP) rounding procedure. Our result, combined with previously known lower bounds, almost settles the integrality gap for a natural LP relaxation. In the second part we consider several well-known clustering problems like k-center, k-median, k-means and their corresponding outlier variants. We use beyond worst-case analysis due to the practical relevance of these problems. In particular we show that when the input instances are 2-perturbation resilient (i.e. the optimal solution does not change when the distances change by a multiplicative factor of 2), the LP integrality gap for k-center (and also asymmetric k-center) is 1. We further introduce a model of perturbation resilience for clustering with outliers. Under this new model, we show that previous results (including our LP integrality result) known for clustering under perturbation resilience also extend for clustering with outliers. This leads to a dynamic programming based heuristic for k-means with outliers (k-means-outlier) which gives an optimal solution when the instance is 2-perturbation resilient. We propose two more algorithms for k-means-outlier — a sampling based algorithm which gives an O(1) approximation when the optimal clusters are not “too small”, and an LP rounding algorithm which gives an O(1) approximation at the expense of violating the number of clusters and outliers by a small constant. We empirically study our proposed algorithms on several clustering datasets

Learning with Submodular Functions: A Convex Optimization Perspective

International audienceSubmodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the lovasz extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, we show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance. By listing many examples of submodular functions, we review various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions

Online Facility Location with Switching Costs

Τα προβλήματα λήψης αποφάσεων αποτελούν μία ευρεία ερευνητική περιοχή και έχουν εξεταστεί σε πολλές διαφορετικές μορφές. Ένας αλγόριθμος για ένα τέτοιο πρόβλημα καλείται να παίρνει αποφάσεις σε γύρους χωρίς να έχει πλήρη γνώση του μέλλοντος και στοχεύει στο να ελαχιστοποιήσει το άθροισμα του κόστους που δέχεται από το περιβάλλον σε όλους τους γύρους, συγκρινόμενος με κάποια συγκεκριμένη ακολουθία αποφάσεων. Ιδιαίτερα ενδιαφέρουσα και επιθυμητή είναι η περίπτωση μία ακολουθίας αποφάσεων που επιτυγχάνει να διατηρεί χαμηλό το συνολικό κόστος που σχετίζεται με την απόφαση κάθε γύρου χωρίς να μεταβάλλεται πολύ στη διάρκεια του χρόνου. Σε αυτή την εργασία περιγράφουμε σημαντικά σημεία της βιβλιογραφίας δύο περιοχών που εξετάζουν αυτά τα προβλήματα και τις διαφορές τους. Τέλος, παρουσιάζουμε έναν πιθανοτικό προσεγγιστικό αλγόριθμο για το πρόβλημα της Άμεσης Χωροθέτησης σε Μεταβαλλόμενο Περιβάλλον που εμπίπτει σε αυτή την κατηγορία.Online decision making is a large research area whose literature includes many different aspects and approaches. The problems it studies are based on the following setting. There is a decision-maker who has to make a decision iteratively with no knowledge of the future and receive the cost of their decision in each round. The goal is to perform well over time. Depending on the definition of what consists of a good performance, that is the benchmark to which we compare our algorithm’s total cost, and on the assumptions made, different kinds of problems occur. A particularly interesting benchmark which captures many real life problems where the environment changes over time, is a solution which balances the trade-off between the optimal costs in each round and its stability. Online learning and competitive analysis are two frameworks which study problems in this setting. In this thesis we will discuss the differences between these two frameworks, the efforts to unify them and finally we will demonstrate how such a unifying approach can give a good approximation algorithm for the online facility location problem with switching costs, which falls into this general setting

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Adaptive Selection Problems in Networked Systems

Networked systems are composed of interconnected nodes that work collaboratively to maximize a given overall utility function. Typical examples of such systems are wireless sensor networks (WSNs) and participatory sensing systems: sensor nodes, either static or mobile, are deployed for monitoring a certain physical field. In these systems, there are a set of problems where we need to adaptively select a strategy to run the system, in order to enhance the efficiency of utilizing the resources available to the system. In particular, we study four adaptive selection problems as follows. We start by studying the problem of base-station (BS) selection in WSNs. Base stations are critical sensor nodes whose failures cause severe data losses. Deploying multiple fixed BSs improves the robustness, yet this scheme is not energy efficient because BSs have high energy consumptions. We propose a scheme that selects only one BS to be active at a time; other BSs are kept passive and act as regular sensor nodes. This scheme substantially reduces the energy supplies required by individual BSs. Then, we propose an algorithm for adaptively selecting the active BS so that the spatially and temporally varying energy resources are efficiently utilized. We also address implementation issues and apply the proposed algorithm on a real WSN. Field experiments have shown the effectiveness of the proposed algorithm. We generalize the BS selection problem by considering both the energy efficiency of regular sensor nodes and that of BSs. In this scheme, a subset of active BSs (instead of only one) is adaptively selected and the routing of regular sensor nodes is adjusted accordingly. Because BSs have high fixed-energy consumptions and because the number of candidate subsets of active BSs is exponential with the number of BSs, this general BS selection problem is NP-hard. We propose a polynomial-time algorithm that is guaranteed, under mild conditions, to achieve a network lifetime at least 62% of the optimal one. Through extensive numerical simulations, we verify that the lifetime achieved by the proposed algorithm is always very close to the optimum. We then study the problem of scheduling the sparse-sensing patterns in WSNs. We observe that the traditional scheme of periodically taking sensing samples is not energy efficient. Instead, we propose to adaptively schedule when and where to activate sensors for sampling a physical field, such that the energy efficiency is enhanced and the sensing precision is maintained. The schedules are learnt from the temporal signal models derived from the collected measurements. Then, using the obtained signal models and the sparse sensing-measurements, the original signal can be effectively recovered. This proposed method requires minimal on-board computation, no inter-node communications and achieves an appealing reconstruction performance. With experiments on real-world datasets, we demonstrate significant improvements over both traditional sensing schemes and the state-of-the-art sparse-sensing schemes, particularly when the measured data is characterized by a strong temporal correlation. In the last part of the thesis, we discuss the sparse-sensing framework by exploiting the spatial correlations rather than the temporal correlations among the captured measurements. In this framework, application-specific utility functions can be employed. By adaptively selecting a small subset of active sensors for sensing, a certain utility is guaranteed and the efficiency of the sensing system is enhanced. We apply this framework both in static WSNs and participatory sensing systems where sensors move in an uncoordinated manner. Through extensive simulations, we show that our proposed algorithm enhances the resource efficiency

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

Fundamentals

Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters