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 factor approximation violating the capacities by a small factor ($1 + ε$) is promising while dealing with capacities.Though LP-rounding has not been very promising while dealing with penalties and outliers, we successfully apply it to deal with them along with capacities. That is, our results are obtained by rounding the solution to the natural LP once again exhibiting the power of LP-rounding technique. Solutions obtained by LP-rounding are easy to integrate with other LP-based algorithms.In this paper, we apply our framework to obtain first constant factor approximations for capacitated facility location problem with outlier (CFLPO) and capacitated $k$-facility location problem with penalty(C$k$FLPP) for hard uniform capacities using LP-rounding. Our solutions incur slight violations in capacities, ($1 + ε$) for the problems without cardinality($k$) constraint and ($2 + ε$) for the problems with the cardinality constraint. For the outlier variant, we also incur a small loss ($1 + ε$) in outliers. Due to the hardness of the underlying problems, the violations are inevitable. Thus we achieve the best possible by rounding the solution of natural LP for these problems. To the best of our knowledge, no results are known for CFLPO and C$k$FLPP.
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…
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…
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…
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 the facility leasing problem with penalties. We present a primal-dual algorithm which is a 3-approximation, based on the algorithm by Nagarajan and Williamson for the facility leasing problem and on the algorithm by Charikar et al. for the facility location problem with penalties.
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 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…
In this paper, we give first constant factor approximation for capacitated knapsack median problem (CKM) for hard uniform capacities, violating the budget only by an additive factor of $f_{max}$ where $f_{max}$ is the maximum cost of a facility opened by the optimal and violating capacities by $(2+ε)$ factor. Natural LP for the problem is known to have an unbounded integrality gap when any one of the two constraints is allowed to be violated by a factor less than $2$. Thus, we present a result which is very close to the best achievable from the natural LP. To the best of our knowledge, the pro…
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…
The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections between clients and opened facilities. We consider the hard-capacitated version of the problem, where a single facility may only serve a limited number of clients and creating multiple copies of a facility is not allowed. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard …
We study problems that integrate depot location decisions along with the inventory routing problem of serving clients from these depots over time balancing the costs of routing vehicles from the depots with the holding costs of demand delivered before they are due. Since the inventory routing problem is already complex, we study the version that assumes that the daily vehicle routes are direct connections from the depot thus forming stars as solutions, and call this problem the Star Inventory Routing Problem with Facility Location (SIRPFL). As a stepping stone to solving SIRPFL, we first study…
In this paper we initiate the study of the heterogeneous capacitated $k$-center problem: given a metric space $X = (F \cup C, d)$, and a collection of capacities. The goal is to open each capacity at a unique facility location in $F$, and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities $c_i$'s are identical, the problem becomes the well-studied uniform capacitated $k$-center problem for which cons…
In this paper, we study locating charging station problem as facility location problem and its variants ($k$-Median, $k$-Facility location and $k$-center). We study the connectivity and the capacity constraints in these problem. Capacity and connectivity constraints have been studied in the literature separately for all these problems. We give first constant factor approximations when both the constraints are present. Extending/modifying the techniques used for connected variants of the problem to include capacities or for capacitated variants of problem to include connectivity is a tedious an…
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…
In the capacitated $k$-median (\CKM) problem, we are given a set $F$ of facilities, each facility $i \in F$ with a capacity $u_i$, a set $C$ of clients, a metric $d$ over $F \cup C$ and an integer $k$. The goal is to open $k$ facilities in $F$ and connect the clients $C$ to the open facilities such that each facility $i$ is connected by at most $u_i$ clients, so as to minimize the total connection cost. In this paper, we give the first constant approximation for \CKM, that only violates the cardinality constraint by a factor of $1+ε$. This generalizes the result of [Li15], which only works for…
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…
The capacitated p-center problem requires to select p facilities from a set of candidates to service a number of customers, subject to facility capacity constraints, with the aim of minimizing the maximum distance between a customer and its associated facility. The problem is well known in the field of facility location, because of the many applications that it can model. In this paper, we solve it by means of search algorithms that iteratively seek the optimal distance by solving tailored subproblems. We present different mathematical formulations for the subproblems and improve them by means…
The $k$-center problem is a classic facility location problem, where given an edge-weighted graph $G = (V,E)$ one is to find a subset of $k$ vertices $S$, such that each vertex in $V$ is "close" to some vertex in $S$. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied. In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor appro…
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 Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Shi Li [SODA'15, SODA'16] gave algorithms violating the number of facilities by a factor of 1+ε exploring properties of extended relaxations. In this paper we develop a constant factor approximation algorithm for Uniform Capacitated k-Median violating only the capacities by a factor of 1+ε. The algorithm is based…
Given facilities with capacities and clients with penalties and demands, the transportation problem with market choice consists in finding the minimum-cost way to partition the clients into unserved clients, paying the penalties, and into served clients, paying the transportation cost to serve them. We give polynomial-time reductions from this problem and variants to the (un)capacitated facility location problem, directly yielding approximation algorithms, two with constant factors in the metric case, one with a logarithmic factor in the general case.
In this paper, we study the uniform capacitated $k$-median problem. Obtaining a constant approximation algorithm for this problem is a notorious open problem; most previous works gave constant approximations by either violating the capacity constraints or the cardinality constraint. Notably, all these algorithms are based on the natural LP-relaxation for the problem. The LP-relaxation has unbounded integrality gap, even when we are allowed to violate the capacity constraints or the cardinality constraint by a factor of $2-ε$. Our result is an $\exp(O(1/ε^2))$-approximation algorithm for the pr…
In this paper, we study the hard uniform capacitated $k$- median problem using local search heuristic. Obtaining a constant factor approximation for the \ckm problem is open. All the existing solutions giving constant-factor approximation, violate at least one of the cardinality and the capacity constraints. All except Koruplou et al are based on LP-relaxation. We give $(3+ε)$ factor approximation algorithm for the problem violating the cardinality by a factor of $8/3 \approx 2.67$. There is a trade-off between the approximation factor and the cardinality violation between our work and the exi…
The state of the art in approximation algorithms for facility location problems are complicated combinations of various techniques. In particular, the currently best 1.488-approximation algorithm for the uncapacitated facility location (UFL) problem by Shi Li is presented as a result of a non-trivial randomization of a certain scaling parameter in the LP-rounding algorithm by Chudak and Shmoys combined with a primal-dual algorithm of Jain et al. In this paper we first give a simple interpretation of this randomization process in terms of solving an aux- iliary (factor revealing) LP. Then, arme…
Capacitated fixed-charge network flows are used to model a variety of problems in telecommunication, facility location, production planning and supply chain management. In this paper, we investigate capacitated path substructures and derive strong and easy-to-compute \emph{path cover and path pack inequalities}. These inequalities are based on an explicit characterization of the submodular inequalities through a fast computation of parametric minimum cuts on a path, and they generalize the well-known flow cover and flow pack inequalities for the single-node relaxations of fixed-charge flow mod…
We study the non-uniform capacitated multi-item lot-sizing (\lotsizing) problem. In this problem, there is a set of demands over a planning horizon of $T$ time periods and all demands must be satisfied on time. We can place an order at the beginning of each period $s$, incurring an ordering cost $K_s$. The total quantity of all products ordered at time $s$ can not exceed a given capacity $C_s$. On the other hand, carrying inventory from time to time incurs inventory holding cost. The goal of the problem is to find a feasible solution that minimizes the sum of ordering and holding costs. Levi e…
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 …
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…
We focus on designing combinatorial algorithms for the Capacitated Network Design problem (Cap-SNDP). The Cap-SNDP is the problem of satisfying connectivity requirements when edges have costs and hard capacities. We begin by showing that the Group Steiner tree problem (GST) is a special case of Cap-SNDP even when there is connectivity requirement between only one source-sink pair. This implies the first poly-logarithmic lower bound for the Cap-SNDP. We next provide combinatorial algorithms for several special cases of this problem. The Cap-SNDP is equivalent to its special case when every edge…
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…
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 …
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…
We consider the {\em matroid median} problem \cite{KrishnaswamyKNSS11}, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation in \cite{KrishnaswamyKNSS11}. Our techniques illustrate that much of the work involved in the…
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 …
We consider the problem of accelerating column generation (CG) for logistics optimization problems using vehicle routing as an example. Without loss of generality, we focus on the Capacitated Vehicle Routing Problem (CVRP) via the addition of a new class of dual optimal inequalities (DOI) that incorporate information about detours from the vehicle routes. These inequalities extend the Smooth-DOI recently introduced in the literature for the solution of certain classes of set-covering problems by CG. The Detour-DOI introduced in this article permit low cost swap operations between items on a gi…
We consider the indirect covering subtree problem (Kim et al., 1996). The input is an edge weighted tree graph along with customers located at the nodes. Each customer is associated with a radius and a penalty. The goal is to locate a tree-shaped facility such that the sum of setup and penalty cost is minimized. The setup cost equals the sum of edge lengths taken by the facility and the penalty cost is the sum of penalties of all customers whose distance to the facility exceeds their radius. The indirect covering subtree problem generalizes the single maximum coverage location problem on trees…
Online algorithms with predictions is a popular and elegant framework for bypassing pessimistic lower bounds in competitive analysis. In this model, online algorithms are supplied with future predictions, and the goal is for the competitive ratio to smoothly interpolate between the best offline and online bounds as a function of the prediction error. In this paper, we study online graph problems with predictions. Our contributions are the following: * The first question is defining prediction error. For graph/metric problems, there can be two types of error, locations that are not predicted, a…
