Approximating soft-capacitated facility location problem with uncertainty

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.     

An improved approximation algorithm for uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the classical facility location problem called uncapacitated facility location problem with penalties (UFLWP for short) in which each client is either assigned to an opened facility or rejected by paying a penalty. The UFLWP problem has been effectively used to model the facility location problem with outliers. Three constant approximation algorithms have been obtained (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001; Jain et al. in J. ACM 50(6):795–824, 2003; Xu and Xu in Inf. Process. Lett. 94(3):119–123, 2005), and the best known performance ratio is 2. The only known hardness result is a 1.463-inapproximability result inherited from the uncapacitated facility location problem (Guha and Khuller in J. Algorithms 31(1):228–248, 1999). In this paper, We present a 1.8526-approximation algorithm for the UFLWP problem. Our algorithm significantly reduces the gap between known performance ratio and the inapproximability result. Our algorithm first enhances the primal-dual method for the UFLWP problem (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001) so that outliers can be recognized more efficiently, and then applies a local search heuristic (Charikar and Guha in Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pp. 378–388, 1999) to further reduce the cost for serving those non-rejected clients. Our algorithm is simple and can be easily implemented. The research of this work was supported in part by NSF through CAREER award CCF-0546509 and grant IIS-0713489. A preliminary version of this paper appeared in the Proceedings of the 11th Annual International Computing and Combinatorics Conference (COCOON’05).     

An approximation algorithm for the <Emphasis Type="Italic">k</Emphasis>-level capacitated facility location problem

We consider the k-level capacitated facility location problem (k-CFLP), which is a natural variant of the classical facility location problem and has applications in supply chain management. We obtain the first (combinatorial) approximation algorithm with a performance factor of \(k+2+\sqrt{k^{2}+2k+5}+\varepsilon\) (ε>0) for this problem.     

An approximation algorithm for <Emphasis Type="Italic">k</Emphasis>-facility location problem with linear penalties using local search scheme

In this paper, we consider an extension of the classical facility location problem, namely k-facility location problem with linear penalties. In contrast to the classical facility location problem, this problem opens no more than k facilities and pays a penalty cost for any non-served client. We present a local search algorithm for this problem with a similar but more technical analysis due to the extra penalty cost, compared to that in Zhang (Theoretical Computer Science 384:126–135, 2007). We show that the approximation ratio of the local search algorithm is \(2 + 1/p + \sqrt{3+ 2/p+ 1/p^2} + \epsilon \), where \(p \in {\mathbb {Z}}_+\) is a parameter of the algorithm and \(\epsilon >0\) is a positive number.     

A local search approximation algorithm for the uniform capacitated <Emphasis Type="Italic">k</Emphasis>-facility location problem

In the uniform capacitated k-facility location problem (UC-k-FLP), we are given a set of facilities and a set of clients. Every client has a demand. Every facility have an opening cost and an uniform capacity. For each client–facility pair, there is an unit service cost to serve the client with unit demand by the facility. The total demands served by a facility cannot exceed the uniform capacity. We want to open at most k facilities to serve all the demands of the clients without violating the capacity constraint such that the total opening and serving cost is minimized. The main contribution of this work is to present the first combinatorial bi-criteria approximation algorithm for the UC-k-FLP by violating the cardinality constraint.     

Line facility location in weighted regions

In this paper, we present approximation algorithms for solving the line facility location problem in weighted regions. The weighted region setup is a more realistic model for many facility location problems that arise in practical applications. Our algorithms exploit an interesting property of the problem, that could possibly be used for solving other problems in weighted regions.     

Approximation algorithm for uniform bounded facility location problem

The uniform bounded facility location problem (UBFLP) seeks for the optimal way of locating facilities to minimize total costs (opening costs plus routing costs), while the maximal routing costs of all clients are at most a given bound M. After building a mixed 0–1 integer programming model for UBFLP, we present the first constant-factor approximation algorithm with an approximation guarantee of 6.853+? for UBFLP on plane, which is composed of the algorithm by Dai and Yu (Theor. Comp. Sci. 410:756–765, 2009) and the schema of Xu and Xu (J. Comb. Optim. 17:424–436, 2008). We also provide a heuristic algorithm based on Benders decomposition to solve UBFLP on general graphes, and the computational experience shows that the heuristic works well.     

Incremental Facility Location Problem and Its Competitive Algorithms

We consider the incremental version of the k-Facility Location Problem, which is a common generalization of the facility location and the k-median problems. The objective is to produce an incremental sequence of facility sets F ₁?F ₂?????F _n , where each F _k contains at most k facilities. An incremental facility sequence or an algorithm producing such a sequence is called c -competitive if the cost of each F _k is at most c times the optimum cost of corresponding k-facility location problem, where c is called competitive ratio. In this paper we present two competitive algorithms for this problem. The first algorithm produces competitive ratio 8α, where α is the approximation ratio of k-facility location problem. By recently result (Zhang, Theor. Comput. Sci. 384:126–135, 2007), we obtain the competitive ratio \(16+8\sqrt{3}+\epsilon\). The second algorithm has the competitive ratio Δ+1, where Δ is the ratio between the maximum and minimum nonzero interpoint distances. The latter result has its self interest, specially for the small metric space with Δ≤8α?1.     

Approximation strategy-proof mechanisms for obnoxious facility location on a line

In the facility location game on a line, there are some agents who have fixed locations on the line where an obnoxious facility will be placed. The objective is to maximize the social welfare, e.g., the sum of distances from the facility to all agents. On collecting location information, agents may misreport the locations so as to stay far away from the obnoxious facility. In this paper, strategy-proof mechanisms are designed and the approximation ratio is used to measure the performances of the strategy-proof mechanisms. Two objective functions, maximizing the sum of squares of distances (maxSOS) and maximizing the sum of distances (maxSum), have been considered. For maxSOS, a randomized 5/3-approximated strategy-proof mechanism is proposed, and the lower bound of the approximation ratio is proved to be at least 1.042. For maxSum, the lower bound of the approximation ratio of the randomized strategy-proof mechanism is proved to be 1.077. Moreover, a general model is considered that each agent may have multiple locations on the line. For the objective functions maxSum and maxSOS, both deterministic and randomized strategy-proof mechanisms are investigated, and the deterministic mechanisms are shown to be best possible.     

The facility location problem with Bernoulli demands

This paper presents the facility location problem with Bernoulli demands. In this capacitated discrete location stochastic problem the goal is to define an a priori solution for the locations of the facilities and for the allocation of customers to the operating facilities that minimizes the sum of the fixed costs of the open facilities plus the expected value of the recourse function. The problem is formulated as a two-stage stochastic program and two different recourse actions are considered. For each of them, a closed form is presented for the recourse function and a deterministic equivalent formulation is obtained for the case in which the probability of demand is the same for all customers. Numerical results from computational experiments are presented and analyzed.     

Online leasing problem with price fluctuations under the consumer price index

We present a \((20+{5}/{n})\)-approximation algorithm for the non-uniform soft capacitated k-facility location problem, violating the capacitated constrains by no more than a factor of 25. The main technique is based on the primal–dual algorithm for the soft capacitated facility location problem, and the exploitation of the combinatorial structure of the fractional solution for the soft capacitated k-facility location problem.     

Strategyproof mechanism design for facility location games with weighted agents on a line

Approximation mechanism design without money was first studied in Procaccia and Tennenholtz (2009) by considering a facility location game. In general, a facility is being opened and the cost of an agent is measured by its distance to the facility. In order to achieve a good social cost, a mechanism selects the location of the facility based on the locations reported by agents. It motivates agents to strategically report their locations to get good outcomes for themselves. A mechanism is called strategyproof if no agents could manipulate to get a better outcome by telling lies regardless of any configuration of other agents. The main contribution in this paper is to explore the strategyproof mechanisms without money when agents are distinguishable. There are two main variations on the nature of agents. One is that agents prefer getting closer to the facility, while the other is that agents prefer being far away from the facility. We first consider the model that directly extends the model in Procaccia and Tennenholtz (2009). In particular, we consider the strategyproof mechanisms without money when agents are weighted. We show that the strategyproof mechanisms in the case of unweighted agents are still the best in the weighted cases. We establish tight lower and upper bounds for approximation ratios on the optimal social utility and the minimum utility when agents prefer to stay close to the facility. We then provide the lower and upper bounds on the optimal social utility and lower bound on the minimum distance per weight when agents prefer to stay far away from the facility. We also extend our study in a natural direction where two facilities must be built on a real line. Secondly, we propose an novel threshold based model to distinguish agents. In this model, we present a strategyproof mechanism that leads to optimal solutions in terms of social cost.     

An approximation algorithm for k-center problem on a convex polygon

This paper studies the constrained version of the k-center location problem. Given a convex polygonal region, every point in the region originates a service demand. Our objective is to place k facilities lying on the region’s boundary, such that every point in that region receives service from its closest facility and the maximum service distance is minimized. This problem is equivalent to covering the polygon by k circles with centers on its boundary which have the smallest possible radius. We present an 1.8841-approximation polynomial time algorithm for this problem.     

Approximation algorithms for <Emphasis Type="Italic">k</Emphasis>-level stochastic facility location problems

In the k-level facility location problem (FLP), we are given a set of facilities, each associated with one of k levels, and a set of clients. We have to connect each client to a chain of opened facilities spanning all levels, minimizing the sum of opening and connection costs. This paper considers the k-level stochastic FLP, with two stages, when the set of clients is only known in the second stage. There is a set of scenarios, each occurring with a given probability. A facility may be opened in any stage, however, the cost of opening a facility in the second stage depends on the realized scenario. The objective is to minimize the expected total cost. For the stage-constrained variant, when clients must be served by facilities opened in the same stage, we present a \((4-o(1))\)-approximation, improving on the 4-approximation by Wang et al. (Oper Res Lett 39(2):160–161, 2011) for each k. In the case with \(k=2,\,3\), the algorithm achieves factors 2.56 and 2.78, resp., which improves the \((3+\epsilon )\)-approximation for \(k=2\) by Wu et al. (Theor Comput Sci 562:213–226, 2015). For the non-stage-constrained version, we give the first approximation for the problem, achieving a factor of 3.495 for the case with \(k = 2\), and \(2k-1+o(1)\) in general.     

A general space-time model for combinatorial optimization problems (and not only)

We consider the problem of defining a strategy consisting of a set of facilities taking into account also the location where they have to be assigned and the time in which they have to be activated. The facilities are evaluated with respect to a set of criteria. The plan has to be devised respecting some constraints related to different aspects of the problem such as precedence restrictions due to the nature of the facilities. Among the constraints, there are some related to the available budget. We consider also the uncertainty related to the performances of the facilities with respect to considered criteria and plurality of stakeholders participating to the decision. The considered problem can be seen as the combination of some prototypical operations research problems: knapsack problem, location problem and project scheduling. Indeed, the basic brick of our model is a variable x_ilt which takes value 1 if facility i is activated in location l at time t, and 0 otherwise. Due to the conjoint consideration of a location and a time in the decision variables, what we propose can be seen as a general space-time model for operations research problems. We discuss how such a model permits to handle complex problems using several methodologies including multiple attribute value theory and multiobjective optimization. With respect to the latter point, without any loss of the generality, we consider the compromise programming and an interactive methodology based on the Dominance-based Rough Set Approach. We illustrate the application of our model with a simple didactic example.     

On recovering syntenic blocks from comparative maps

A genomic map is represented by a sequence of gene markers, and a gene marker can appear in several different genomic maps, in either positive or negative form. A strip (syntenic block) is a sequence of distinct markers that appears as subsequences in two or more maps, either directly or in reversed and negated form. Given two genomic maps G and H, the problem Maximal Strip Recovery (MSR) is to find two subsequences G′ and H′ of G and H, respectively, such that the total length of disjoint strips in G′ and H′ is maximized. Previously only a heuristic was provided for this problem, which does not guarantee finding the optimal solution, and it was unknown whether the problem is NP-hard or polynomially solvable. In this paper, we develop a factor-4 polynomial-time approximation algorithm for the problem, and show that several close variants of the problem are intractable.     

Algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP₂) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algorithm in time O(NlogN) for finding a minimum weight non-separating path between two given nodes in a grid graph of N nodes with positive weight. This result leads to a 5/4-approximation algorithm for the BCP₂ problem on grid graphs, which is the currently best ratio achieved in polynomial time. We also developed an exact algorithm for the BCP₂ problem on grid graphs. Based on the exact algorithm and a rounding technique, we show an approximation scheme, which is a fully polynomial time approximation scheme for fixed number of rows.     

Median problems with positive and negative weights on cycles and cacti

This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest facility. For the MWD model it is shown that there exists a finite set of points in the graph which contains the locations of facilities in an optimal solution. Furthermore, algorithms for both models for the 2-median problem on a cycle are developed. The algorithm for the MWD model runs in linear time, whereas the algorithm for the WMD model has a time complexity of  O(n²)\mathcal{O}(n^{2}) .     

Facility Location with Dynamic Distance Functions

Facility location problems have always been studied with theassumption that the edge lengths in the network are static anddo not change over time. The underlying network could be used to model a city street networkfor emergency facility location/hospitals, or an electronic network for locating information centers. In any case, it is clear that due to trafficcongestion the traversal time on links changes with time. Very often, we have estimates as to how the edge lengths change over time, and our objective is to choose a set of locations (vertices) ascenters, such that at every time instant each vertex has a center close to it (clearly, the center close to a vertex may change over time). We also provide approximation algorithms as well as hardness results forthe K-center problem under this model. This is the first comprehensive study regarding approximation algorithmsfor facility location for good time-invariant solutions.     

A tighter formulation of the <Emphasis Type="Italic">p</Emphasis>-median problem

Given a set of clients and a set of potential sites for facilities, the p-median problem consists of opening a set of p sites and assigning each client to the closest open facility to it. It can be viewed as a variation of the uncapacitated facility location problem. We propose a new formulation of this problem by a mixed integer linear problem. We show that this formulation, while it has the same value by LP-relaxation, can be much more efficient than two previous formulations. The computational experiment performed on two sets of benchmark instances has showed that the efficiency of the standard branch-and-cut algorithm has been significantly improved. Finally, we explore the structure of the new formulation in order to derive reduction rules and to accelerate the LP-relaxation resolution.