We study local search algorithms for metric instances of facility location problems: the uncapacitated facility location problem (UFL), as well as uncapacitated versions of the $k$-median, $k$-center and $k$-means problems. All these problems admit natural local search heuristics: for example, in the UFL problem the natural moves are to open a new facility, close an existing facility, and to swap a closed facility for an open one; in $k$-medians, we are allowed only swap moves. The local-search algorithm for $k$-median was analyzed by Arya et al. (SIAM J. Comput. 33(3):544-562, 2004), who used a clever ``coupling'' argument to show that local optima had cost at most constant times the global optimum. They also used this argument to show that the local search algorithm for UFL was 3-approximation; their techniques have since been applied to other facility location problems.
In this paper, we give a proof of the $k$-median result which avoids this coupling argument. These arguments can be used in other settings where the Arya et al. arguments have been used. We also show that for the problem of opening $k$ facilities $F$ to minimize the objective function $Φ_p(F) = \big(\sum_{j \in V} d(j, F)^p\big)^{1/p}$, the natural swap-based local-search algorithm is a $Θ(p)$-approximation. This implies constant-factor approximations for $k$-medians (when $p=1$), and $k$-means (when $p = 2$), and an $O(\log n)$-approximation algorithm for the $k$-center problem (which is essentially $p = \log n$).
We describe the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs, and for (non-metric) fractional facility location. We also describe the first parallel algorithms requiring only near-linear total work and finishing in polylog time. The algorithms compute $(1+ε)$-approximate solutions in time (and work) $O^*(N/ε^2)$, where $N$ is the number of non-zeros in the constraint matrix. For facility location, $N$ is the number of eligible client/facility pairs.
We first show that a better analysis of the algorithm for The Two-Sage Stochastic Facility Location Problem from Srinivasan \cite{sri07} and the algorithm for The Robust Fault Tolerant Facility Location Problem from Byrka et al \cite{bgs10} can render improved approximation factors of 2.206 and α+4 where αis the maximum number an adversary can close, respectively, and which are the best ratios so far. We then present new models for the soft-capacitated facility location problem with uncertainty and design constant factor approximation algorithms to solve them. We devise the stochastic and robu…
Facility Location problems ask to place facilities in a way that optimizes a given objective function so as to provide a service to all clients. These are one of the most well-studied optimization problems spanning many research areas such as operations research, computer science, and management science. Traditionally, these problems are solved with the assumption that clients need to be served by one facility each. In many real-world scenarios, it is very likely that clients need a robust service that requires more than one facility for each client. In this paper, we capture this robustness b…
We study a variant of the uncapacitated facility location problem (UFL), where connection costs of clients are defined by (client specific) concave nondecreasing functions of the connection distance in the underlying metric. A special case capturing the complexity of this variant is the setting called facility location with penalties where clients may either connect to a facility or pay a (client specific) penalty. We show that the best known approximation algorithms for UFL may be adapted to the concave connection cost setting. The key technical contribution is an argument that the JMS algori…
The metric capacitated facility location is a well-studied problem for which, while constant factor approximations are known, no efficient relaxation with constant integrality gap is known. The question whether there is such a relaxation is among the most important open problems of approximation algorithms \cite{ShmoysWbook}. In this paper we show that, if one is restricted to linear programs that use the natural encoding for facility location, at least an exponential number of constraints is needed to achieve a constant gap. Our proof does not assume any special property of the relaxation suc…
The \emph{dynamic facility location problem} is a generalization of the classic facility location problem proposed by Eisenstat, Mathieu, and Schabanel to model the dynamics of evolving social/infrastructure networks. The generalization lies in that the distance metric between clients and facilities changes over time. This leads to a trade-off between optimizing the classic objective function and the "stability" of the solution: there is a switching cost charged every time a client changes the facility to which it is connected. While the standard linear program (LP) relaxation for the classic …
We study the metric facility location problem with client insertions and deletions. This setting differs from the classic dynamic facility location problem, where the set of clients remains the same, but the metric space can change over time. We show a deterministic algorithm that maintains a constant factor approximation to the optimal solution in worst-case time $\tilde O(2^{O(κ^2)})$ per client insertion or deletion in metric spaces while answering queries about the cost in $O(1)$ time, where $κ$ denotes the doubling dimension of the metric. For metric spaces with bounded doubling dimension…
We consider the classic facility location problem in fully dynamic data streams, where elements can be both inserted and deleted. In this problem, one is interested in maintaining a stable and high quality solution throughout the data stream while using only little time per update (insertion or deletion). We study the problem and provide the first algorithm that at the same time maintains a constant approximation and incurs polylogarithmic amortized recourse per update. We complement our theoretical results with an experimental analysis showing the practical efficiency of our method.
We consider the uncapacitated facility location problem with (linear) penalty function and show that a modified JMS algorithm, combined with a randomized LP rounding technique due to Byrka-Aardal[1], Li[14] and Li et al.[16] yields 1.488 approximation, improving the factor 1.5148 due to Li et al.[16]. This closes the current gap between the classical facility location problem and this penalized variant. Main ingredient is a straightforward adaptation of the JMS algorithm to the penalty setting plus a consistent use of the upper bounding technique for factor revealing LPs due to Fernandes et al…
Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees we…
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's location. However, when a client who has been also one of the open facilities is to be removed, then…
In this paper, we give the first constant approximation algorithm for the lower bounded facility location (LBFL) problem with general lower bounds. Prior to our work, such algorithms were only known for the special case where all facilities have the same lower bound: Svitkina \cite{Svi10} gave a $448$-approximation for the special case, and subsequently Ahmadian and Swamy \cite{AS13} improved the approximation factor to 82.6. As in \cite{Svi10} and \cite{AS13}, our algorithm for LBFL with general lower bounds works by reducing the problem to the capacitated facility location (CFL) problem. To …
In this work, we study the extension of two variants of the facility location problem (FL) to make them robust towards a few distantly located clients. First, $k$-facility location problem ($k$FL), a common generalization of FL and $k$ median problems, is a well studied problem in literature. In the second variant, lower bounded facility location (LBFL), we are given a bound on the minimum number of clients that an opened facility must serve. Lower bounds are required in many applications like profitability in commerce and load balancing in transportation problem. In both the cases, the cost o…
The uncapacitated facility location has always been an important problem due to its connection to operational research and infrastructure planning. Byrka obtained an algorithm that is parametrized by $γ$ and proved that it is optimal when $γ>1.6774$. He also proved that the algorithm achieved an approximation ratio of 1.50. A later work by Shi Li achieved an approximation factor of 1.488. In this research, we studied these algorithms and several related works. Although we didn't improve upon the algorithm of Shi Li, our work did provide some insight into the problem. We also reframed the probl…
We consider the classic Facility Location problem on planar graphs (non-uniform, uncapacitated). Given an edge-weighted planar graph $G$, a set of clients $C\subseteq V(G)$, a set of facilities $F\subseteq V(G)$, and opening costs $\mathsf{open} \colon F \to \mathbb{R}_{\geq 0}$, the goal is to find a subset $D$ of $F$ that minimizes $\sum_{c \in C} \min_{f \in D} \mathrm{dist}(c,f) + \sum_{f \in D} \mathsf{open}(f)$. The Facility Location problem remains one of the most classic and fundamental optimization problem for which it is not known whether it admits a polynomial-time approximation sch…
In the uncapacitated facility location problem, given a graph, a set of demands and opening costs, it is required to find a set of facilities R, so as to minimize the sum of the cost of opening the facilities in R and the cost of assigning all node demands to open facilities. This paper concerns the robust fault-tolerant version of the uncapacitated facility location problem (RFTFL). In this problem, one or more facilities might fail, and each demand should be supplied by the closest open facility that did not fail. It is required to find a set of facilities R, so as to minimize the sum of the…
Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for \cfl\ is among the top 10 …
In this paper, we address the problem of capacitated facility location problem with penalties (CapFLPP) paid per unit of unserved demand. In case of uncapacitated FLP with penalties demands of a client are either entirely met or are entirely rejected and penalty is paid. In the uncapacitated case, there is no reason to serve a client partially. Whereas, in case of CapFLPP, it may be beneficial to serve a client partially instead of not serving at all and, pay the penalty for the unmet demand. Charikar et. al. \cite{charikar2001algorithms}, Jain et. al. \cite{jain2003greedy} and Xu- Xu \cite{xu…
Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under various settings reflecting typical structures within the optimization problems of practical interest, little is known on how the problem behaves in conjunction with parity constraints. This shortfall of understanding was rather disturbing when we consider the central role of parity in the field of combinatorics. In this paper, we present the first constant-factor approximation algorith…
The metric uncapacitated facility location problem (UFL) enjoys a special stature in approximation algorithms as a testbed for various techniques. Two generalizations of UFL are capacitated facility location (CFL) and lower-bounded facility location (LBFL). In the former, every facility has a capacity which is the maximum demand that can be assigned to it, while in the latter, every open facility is required to serve a given minimum amount of demand. Both CFL and LBFL are approximable within a constant factor but their respective natural LP relaxations have an unbounded integrality gap. Accord…
We provide nearly optimal algorithms for online facility location (OFL) with predictions. In OFL, $n$ demand points arrive in order and the algorithm must irrevocably assign each demand point to an open facility upon its arrival. The objective is to minimize the total connection costs from demand points to assigned facilities plus the facility opening cost. We further assume the algorithm is additionally given for each demand point $x_i$ a natural prediction $f_{x_i}^{\mathrm{pred}}$ which is supposed to be the facility $f_{x_i}^{\mathrm{opt}}$ that serves $x_i$ in the offline optimal solution…
We present a simple, robust and efficient harmony search algorithm for the Hop Constrained Connected Facility Location problem (HCConFL). The HCConFL problem is NP-hard that models the design of data-management and telecommunication networks in a manner of reliability. In this paper, we customize harmony search algorithm to solve the HCConFL problem. To arrive to quick, optimal cost of each solution, we use a new greedy approach expanding idea of Kruskal algorithm in our objective function. We also use a new greedy method combined with harmony search to obtain a good approximation in an effici…
In this paper, we present a framework to design approximation algorithms for capacitated facility location problems with penalties/outliers using LP-rounding. Primal-dual technique, which has been particularly successful in dealing with outliers and penalties, has not been very successful in dealing with capacities. On the other hand, no primal-dual solution has been able to break the hardness of capacitated facility location problem(CFLP). LP-rounding techniques had also not been very successful in dealing with capacities until a recent work by Grover et al. \cite{GroverGKP18}. Their constant…
The Capacitated Facility Location (CFL), a long-standing classic problem with intriguing approximability and literature dated back to the 90s, is considered. Following the open question posted in [Williamson and Shmoys, 2011] and the notable work due to [An et al., FOCS~2014], we present an LP-based approximation algorithm with a guarantee of $(10+\sqrt{67})/2 \approx 9.0927$, a significant improvement upon the previous LP-based ratio of $288$ due to An et al. in 2014. Our contribution for this part is a simple and elegant rounding algorithm that brings clear insights for the MFN relaxation an…
In this paper we study the facility location problem in the online with recourse and dynamic algorithm models. In the online with recourse model, clients arrive one by one and our algorithm needs to maintain good solutions at all time steps with only a few changes to the previously made decisions (called recourse). We show that the classic local search technique can lead to a $(1+\sqrt{2}+ε)$-competitive online algorithm for facility location with only $O\left(\frac{\log n}ε\log\frac1ε\right)$ amortized facility and client recourse. We then turn to the dynamic algorithm model for the problem, …
We study the capacitated $k$-facility location problem, in which we are given a set of clients with demands, a set of facilities with capacities and a constant number $k$. It costs $f_i$ to open facility $i$, and $c_{ij}$ for facility $i$ to serve one unit of demand from client $j$. The objective is to open at most $k$ facilities serving all the demands and satisfying the capacity constraints while minimizing the sum of service and opening costs. In this paper, we give the first fully polynomial time approximation scheme (FPTAS) for the single-sink (single-client) capacitated $k$-facility loca…
We give a new randomized LP-rounding 1.725-approximation algorithm for the metric Fault-Tolerant Uncapacitated Facility Location problem. This improves on the previously best known 2.076-approximation algorithm of Swamy & Shmoys. To the best of our knowledge, our work provides the first application of a dependent-rounding technique in the domain of facility location. The analysis of our algorithm benefits from, and extends, methods developed for Uncapacitated Facility Location; it also helps uncover new properties of the dependent-rounding approach. An important concept that we develop is a no…
In this note, we consider the capacitated facility location problem when the transportation costs of the instance satisfy the Monge property. We show that a straightforward dynamic program finds the optimal solution when the demands are polynomially bounded. When demands are not polynomially bounded, we give a fully polynomial-time approximation scheme by adapting an algorithm and analysis of Van Hoesel and Wagelmans.
We study the problem of online facility location with delay. In this problem, 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 penalty: each client incurs a waiting cost (the difference between its arrival and connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. This is a well-studied problem both of its own, and within the general class of network design problems with dela…
In the online non-metric variant of the facility location problem, there is a given graph consisting of a set $F$ of facilities (each with a certain opening cost), a set $C$ of potential clients, and weighted connections between them. The online part of the input is a sequence of clients from $C$, and in response to any requested client, an online algorithm may open an additional subset of facilities and must connect the given client to an open facility. We give an online, polynomial-time deterministic algorithm for this problem, with a competitive ratio of $O(\log |F| \cdot (\log |C| + \log \…
A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor of an associated algorithm. Obtaining an UPFRP is straightforward, and can be used as an alternative to analytical proofs, that are usually very long and tedious. We apply this technique to the Metric Facility Location Problem (MFLP) and to a generalization where the distance function is a squared metric. We call this generalization the Squared…
This paper presents a distributed O(1)-approximation algorithm, with expected-$O(\log \log n)$ running time, in the $\mathcal{CONGEST}$ model for the metric facility location problem on a size-$n$ clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric …
We give a concentration inequality for a stochastic version of the facility location problem. We show the objective $C_n = \min_{F \subseteq [0,1]^2}|F|+\sum_{x\in X}\min_{f\in F}\|x-f\|$ is concentrated in an interval of length $O(n^{1/6})$ and $\E[C_n]=Θ(n^{2/3})$ if the input $X$ consists of i.i.d. uniform points in the unit square. Our main tool is to use a geometric quantity, previously used in the design of approximation algorithms for the facility location problem, to analyze a martingale process. Many of our techniques generalize to other settings.
We consider the {\em lower-bounded facility location} (\lbfl) problem (also sometimes called {\em load-balanced facility location}), which is a generalization of {\em uncapacitated facility location} (\ufl), where each open facility is required to serve a certain {\em minimum} amount of demand. More formally, an instance $\I$ of \lbfl is specified by a set $\F$ of facilities with facility-opening costs $\{f_i\}$, a set $\D$ of clients, and connection costs $\{c_{ij}\}$ specifying the cost of assigning a client $j$ to a facility $i$, where the $c_{ij}$s form a metric. A feasible solution specif…
The facility location problem is an NP-hard optimization problem. Therefore, approximation algorithms are often used to solve large instances. Such algorithms often perform much better than worst-case analysis suggests. Therefore, probabilistic analysis is a widely used tool to analyze such algorithms. Most research on probabilistic analysis of NP-hard optimization problems involving metric spaces, such as the facility location problem, has been focused on Euclidean instances, and also instances with independent (random) edge lengths, which are non-metric, have been researched. We would like t…
We study different restricted variations of the obnoxious facility location problem on a plane. The first is the constrained obnoxious facility location on a line segment (COFL-Line) problem. We provide an efficient algorithm for this problem that executes in $O(n ^ 2 \log k + n \log k \log (n^2 + k))$ time. Our result improves on the best known result of $O((nk)^2 \log(nk) + (n + k) \log (nk))$ time obtained by Singireddy and Basappa\cite{singireddy2022dispersing}. We also study the same problem where the facilities must be placed on a given circle (the constrained obnoxious facility location…
This paper presents fast, distributed, $O(1)$-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the $k$-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an $n \times n$ matrix distributed among the machines) or implicitly as …
In the Submodular Facility Location problem (SFL) we are given a collection of $n$ clients and $m$ facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility $f$ to which we assign the subset of clients $S^f$, one has to pay the opening cost $g(S^f)$, where $g(\cdot)$ is a monotone submodular function with $g(\emptyset)=0$. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility …
Improved Bounds for Online Facility Location with Predictions
National Technical University of Athens（雅典国立技术大学）；Archimedes/Athena RC（阿基米德/雅典娜研究中心）；University of Wisconsin - Madison（威斯康星大学麦迪逊分校）；National University of Singapore（新加坡国立大学）；University of California, Irvine（加州大学欧文分校）
In this paper, we will formalize the method of dual fitting and the idea of factor-revealing LP. This combination is used to design and analyze two greedy algorithms for the metric uncapacitated facility location problem. Their approximation factors are 1.861 and 1.61, with running times of O(mlog m) and O(n^3), respectively, where n is the total number of vertices and m is the number of edges in the underlying complete bipartite graph between cities and facilities. The algorithms are used to improve recent results for several variants of the problem.
This paper presents the design and analysis of parallel approximation algorithms for facility-location problems, including $\NC$ and $\RNC$ algorithms for (metric) facility location, $k$-center, $k$-median, and $k$-means. These problems have received considerable attention during the past decades from the approximation algorithms community, concentrating primarily on improving the approximation guarantees. In this paper, we ask, is it possible to parallelize some of the beautiful results from the sequential setting? Our starting point is a small, but diverse, subset of results in approximation…
Understanding the dynamics of evolving social or infrastructure networks is a challenge in applied areas such as epidemiology, viral marketing, or urban planning. During the past decade, data has been collected on such networks but has yet to be fully analyzed. We propose to use information on the dynamics of the data to find stable partitions of the network into groups. For that purpose, we introduce a time-dependent, dynamic version of the facility location problem, that includes a switching cost when a client's assignment changes from one facility to another. This might provide a better rep…
With growing emphasis on e-commerce marketplace platforms where we have a central platform mediating between the seller and the buyer, it becomes important to keep a check on the availability and profitability of the central store. A store serving too less clients can be non-profitable and a store getting too many orders can lead to bad service to the customers which can be detrimental for the business. In this paper, we study the facility location problem(FL) with upper and lower bounds on the number of clients an open facility serves. Constant factor approximations are known for the restrict…
The paper gives approximation algorithms for the k-medians and facility-location problems (both NP-hard). For k-medians, the algorithm returns a solution using at most ln(n+n/epsilon)k medians and having cost at most (1+epsilon) times the cost of the best solution that uses at most k medians. Here epsilon > 0 is an input to the algorithm. In comparison, the best previous algorithm (Jyh-Han Lin and Jeff Vitter, 1992) had a (1+1/epsilon)ln(n) term instead of the ln(n+n/epsilon) term in the performance guarantee. For facility location, the algorithm returns a solution of cost at most d+ln(n) k, p…
The study of approximate mechanism design for facility location problems has been in the center of research at the intersection of artificial intelligence and economics for the last decades, largely due to its practical importance in various domains, such as social planning and clustering. At a high level, the goal is to design mechanisms to select a set of locations on which to build a set of facilities, aiming to optimize some social objective and ensure desirable properties based on the preferences of strategic agents, who might have incentives to misreport their private information such as…
We consider the {\em mobile facility location} (\mfl) problem. We are given a set of facilities and clients located in a common metric space. The goal is to move each facility from its initial location to a destination and assign each client to the destination of some facility so as to minimize the sum of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We …
Logistics network is expected that opened facilities work continuously for a long time horizon without any failure, but in real world problems, facilities may face disruptions. This paper studies a reliable joint inventory location problem to optimize the cost of facility locations, customers assignment, and inventory management decisions when facilities face failure risks and do not work. In our model we assume when a facility is out of work, its customers may be reassigned to other operational facilities otherwise they must endure high penalty costs associated with losing service. For defini…
We study the multistage $K$-facility reallocation problem on the real line, where we maintain $K$ facility locations over $T$ stages, based on the stage-dependent locations of $n$ agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. $K$-facility reallocation was introduced by de Keijzer and Wojtczak, where they mostly focused on the special case of a single…
SUNY at Buffalo（纽约州立大学布法罗分校）；Boston University（波士顿大学）
In this paper we study the uncapacitated facility location problem in the model of differential privacy (DP) with uniform facility cost. Specifically, we first show that, under the hierarchically well-separated tree (HST) metrics and the super-set output setting that was introduced in Gupta et. al., there is an $ε$-DP algorithm that achieves an $O(\frac{1}ε)$(expected multiplicative) approximation ratio; this implies an $O(\frac{\log n}ε)$ approximation ratio for the general metric case, where $n$ is the size of the input metric. These bounds improve the best-known results given by Gupta et. a…
Recent years have witnessed the rise of many successful e-commerce marketplace platforms like the Amazon marketplace, AirBnB, Uber/Lyft, and Upwork, where a central platform mediates economic transactions between buyers and sellers. Motivated by these platforms, we formulate a set of facility location problems that we term Two-sided Facility location. In our model, agents arrive at nodes in an underlying metric space, where the metric distance between any buyer and seller captures the quality of the corresponding match. The platform posts prices and wages at the nodes, and opens a set of facil…
Massachusetts Institute of Technology（麻省理工学院）；Institute of Advanced Study（高等研究院）
Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-linkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous stat…
