Online Facility Location with Deletions

In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment. We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client\u27s location. However, when a client who has been also one of the open facilities is to be removed, then our model has to allow to reconnect all clients that have been connected to that removed facility. In this model, we present an optimal O(log(n_{act}) / log log(n_{act}))-competitive algorithm, where n_{act} is the number of active clients at the end of the input sequence. Next, we turn our attention to the capacitated Facility Location problem. We first note that if no deletions are allowed, then one can achieve an optimal competitive ratio of O(log(n) / log(log n)), where n is the length of the sequence. However, when deletions are allowed, the capacitated version of the problem is significantly more challenging than the uncapacitated one. We show that still, using a more sophisticated algorithmic approach, one can obtain an online O(log N + log c log n)-competitive algorithm for the capacitated Facility Location problem in the fully dynamic model, where N is number of points in the input metric and c is the capacity of any open facility

Learning Augmented Online Facility Location

Following the research agenda initiated by Munoz & Vassilvitskii [1] and Lykouris & Vassilvitskii [2] on learning-augmented online algorithms for classical online optimization problems, in this work, we consider the Online Facility Location problem under this framework. In Online Facility Location (OFL), demands arrive one-by-one in a metric space and must be (irrevocably) assigned to an open facility upon arrival, without any knowledge about future demands. We present an online algorithm for OFL that exploits potentially imperfect predictions on the locations of the optimal facilities. We prove that the competitive ratio decreases smoothly from sublogarithmic in the number of demands to constant, as the error, i.e., the total distance of the predicted locations to the optimal facility locations, decreases towards zero. We complement our analysis with a matching lower bound establishing that the dependence of the algorithm's competitive ratio on the error is optimal, up to constant factors. Finally, we evaluate our algorithm on real world data and compare our learning augmented approach with the current best online algorithm for the problem

Online Facility Location with Linear Delay

In the problem of online facility location with delay, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a certain penalty: each client incurs a waiting cost (equal to the difference between its arrival and its connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. That is, an algorithm needs to balance three types of costs: cost of opening facilities, costs of connecting clients, and the waiting costs of clients. We study a natural variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay, where clients may connect to a facility only at its opening time. We present an O(1)-competitive deterministic algorithm for the two-sided delay variant. Our approach is an extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location. To this end, we substantially simplify the part of the original argument in which a bound on the sequence of factor-revealing LPs is derived. We then show how to transform our O(1)-competitive algorithm for the two-sided delay variant to O(log n / log log n)-competitive deterministic algorithm for one-sided delays. This improves the known O(log n) bound by Azar and Touitou [FOCS 2020]. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios

Online Facility Location with Linear Delay

In the problem of online facility location with delay, a sequence of n clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a certain penalty: each client incurs a waiting cost (equal to the difference between its arrival and its connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. That is, an algorithm needs to balance three types of costs: cost of opening facilities, costs of connecting clients, and the waiting costs of clients. We study a natural variant of this problem, where clients may be connected also to an already open facility, but such action incurs an extra cost: an algorithm pays for waiting of the facility (a cost incurred separately for each such "late" connection). This is reminiscent of online matching with delays, where both sides of the connection incur a waiting cost. We call this variant two-sided delay to differentiate it from the previously studied one-sided delay, where clients may connect to a facility only at its opening time. We present an O(1)-competitive deterministic algorithm for the two-sided delay variant. Our approach is an extension of the approach used by Jain, Mahdian and Saberi [STOC 2002] for analyzing the performance of offline algorithms for facility location. To this end, we substantially simplify the part of the original argument in which a bound on the sequence of factor-revealing LPs is derived. We then show how to transform our O(1)-competitive algorithm for the two-sided delay variant to O(log n / log log n)-competitive deterministic algorithm for one-sided delays. This improves the known O(log n) bound by Azar and Touitou [FOCS 2020]. We note that all previous online algorithms for problems with delays in general metrics have at least logarithmic ratios

Online facility location with timed-requests and congestion

The classic online facility location problem deals with finding the optimal set of facilities in an online fashion when demand requests arrive one at a time and facilities need to be opened to service these requests. In this work, we study two variants of the online facility location problem; (1) timed requests and (2) congestion. Both of these variants are motivated by the applications to real life and the previously known results on online facility location cannot be directly adapted to analyse them. Timed requests : In this variant, each demand request is a pair $(x,t)$ where the $x$ is the standard location of the demand while $t$ is the corresponding weight of the request. The cost of servicing request $(x,t)$ at facility $F$ is $t\cdot d(x,F')$ where $F'$ is the set of facilities available at the time of request $(x,t)$. For this variant, we present an online algorithm attaining a competitive ratio of $\mathcal{O}(\log n)$ in the secretarial model for the timed requests and show that it is optimal. Congestion : The congestion variant considers the case when there is an additional congestion cost that grows with the number of requests served by each request. For this variant, when the congestion cost is a monomial, we show that there exists an algorithm attaining a constant competitive ratio. This constant is a function of the exponent of the monomial and the facility opening cost but independent of the number of requests.Comment: 25 pages, 6 figure

Submodular Norms with Applications To Online Facility Location and Stochastic Probing

Optimization problems often involve vector norms, which has led to extensive research on developing algorithms that can handle objectives beyond ?_p norms. Our work introduces the concept of submodular norms, which are a versatile type of norms that possess marginal properties similar to submodular set functions. We show that submodular norms can either accurately represent or approximate well-known classes of norms, such as ?_p norms, ordered norms, and symmetric norms. Furthermore, we establish that submodular norms can be applied to optimization problems such as online facility location and stochastic probing. This allows us to develop a logarithmic-competitive algorithm for online facility location with symmetric norms, and to prove logarithmic adaptivity gap for stochastic probing with symmetric norms

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