P-center Problem Research Articles

Approximating the probabilistic p-Center problem under pressure

The Probabilistic p-Center problem under Pressure (Min PpCP) is a variant of the usual Minp-Center problem we recently introduced in the context of wildfire management. The problem is to locate p shelters minimizing the maximum distance people will have to cover in case of fire in order to reach the closest accessible shelter. The landscape is divided into zones and is modeled as an edge-weighted graph with vertices corresponding to zones and edges corresponding to direct connections between two adjacent zones. The risk associated with fire outbreaks is modeled using a finite set of fire scenarios. Each scenario corresponds to a fire outbreak on a single zone (i.e., on a vertex) with the main consequence of modifying evacuation paths in two ways. First, an evacuation path cannot pass through the vertex on fire. Second, the fact that someone close to the fire may not take rational decisions when selecting a direction to escape is modeled using new kinds of evacuation paths. In this paper, we characterize the set of feasible solutions of Min PpCP-instance. Then, we propose some approximation results for Min PpCP. These results require approximation results for two variants of the (deterministic) Minp-Center problem called Min MACp-Center and Min Partialp-Center.

A parallel variable neighborhood search for [formula omitted]-neighbor facility location problems

In this paper, we employ the less is more approach to develop a Parallel Variable Neighborhood Search (VNS) algorithm for the α-neighbor p-center problem (αNpCP) and the α-neighbor p-median problem (αNpMP). The αNpCP and the αNpMP are generalizations of the p-center (pCP) and p-median (pMP) problems, respectively. In the α-neighbor problems, one seeks to open p facilities and assign each of the n customers to their closest α ones. The objective is to minimize the maximum distance of a customer to its αth facility, in the case of the αNpCP, and the sum of the distances from each customer to their α nearest facilities, in the case of the αNpMP. Our VNS adapts simple but efficient algorithms and data structures from the pCP and pMP literature to the αNpCP and αNpMP context. We also introduce an updated objective function for the αNpCP, which adds more information to the solution cost and helps the VNS to escape from local optima. Several experimental tests show that our VNS outperforms more complex state-of-the-art algorithms. Regarding the αNpCP, on 120 instances derived from the OR-library set, our algorithm improved best-known solutions for 22, with an average improvement of 34.26%; the overall gap on the 120 instances is 6.18% in favor of our algorithm. Moreover, on 231 instances derived from the TSPLIB set, we improved the solutions for 115, with an average improvement of 5.30%, and an overall improvement gap of 2.47% for all 231 instances. Considering the αNpMP results, our heuristic obtained better results than a heuristic from literature in all 80 instances tested, finding optimal solutions in all these instances.

A vertex weighting-based double-tabu search algorithm for the classical [formula omitted]-center problem

The p-center problem, which is NP-hard, aims to select p centers from a set of candidates to serve all clients while minimizing the maximum distance between each client and its assigned center. To solve this challenging optimization problem, we transform the p-center problem into a series of decision subproblems, and propose a vertex weighting-based double-tabu search (VWDT) algorithm. It integrates a vertex weighting strategy and a double-tabu search which combines both solution-based and attribute-based tabu strategies to help the search to escape from the local optima trap. Computational experiments on totally 510 public instances in the literature show that VWDT is highly competitive comparing to the state-of-the-art algorithms. Specifically, VWDT improves the previous best known results for 84 large instances and matches the best results for all the remaining ones. Apart from the improvements in solution quality, VWDT is much faster than other state-of-the-art algorithms in the literature, especially on some large instances. Furthermore, we perform additional experiments to analyze the impact of the key components to VWDT, such as the vertex weighting and the double-tabu search strategy.

Robust MILP formulations for the two-stage weighted vertex [formula omitted]-center problem

The weighted vertex p-center problem (PCP) consists of locating p facilities among a set of available sites such that the maximum weighted distance (or travel time) from any demand node to its closest located facility is minimized. This paper studies the exact solution of the two-stage robust weighted vertex p-center problem (RPCP2). In this problem, the location of the facilities is fixed in the first stage while the demand node allocations are recourse decisions fixed once the uncertainty is revealed. The problem is modeled by box uncertainty sets on both the demands and the distances. We introduce five different robust reformulations based on MILP formulations of (PCP) from the literature. We prove that considering a finite subset of scenarios is sufficient to obtain an optimal solution of (RPCP2). We leverage this result to introduce a column-and-constraint generation algorithm and a branch-and-cut algorithm to efficiently solve this problem optimally. We highlight how these algorithms can also be adapted to solve the single-stage problem (RPCP1) which is obtained when no recourse is considered. We present a numerical study to compare the performances of these formulations on randomly generated instances and on a case study from the literature.

A parallel mayfly algorithm for the [formula omitted]-neighbor p-center problem

This study presents a competitive algorithm in which the mayfly optimization algorithm (MA) is combined with a local search procedure to provide high-quality solutions to the α-neighbor p-center problem (α – pCP). The aim of this problem is to locate p facilities on a plane such that the maximum distance between each demand point and its αth closest facility is minimized. Although α – pCP is a discrete optimization problem, it is addressed in this study from the perspective of a continuous optimization problem. The major reason for this choice is that the MA is a very suitable technique for finding the optimum point on a plane. However, it is very difficult to successfully apply this technique to a discrete optimization problem. Hence, additional efforts are made to reduce the computational burden that arises from approaching this from the perspective of a continuous problem. Thus, an effective solution method based on the MA for finding the optimum point on a plane is presented for the discrete α – pCP. The proposed MA method is compared to the best alternatives found in the literature, which are an extremely efficient exact procedure for the continuous variant of the problem, and a GRASP algorithm that includes a tabu search with strategic oscillation (SO) post-processing. A comparative analysis shows that the performance of the proposed algorithm is similar to that of these state-of-the-art methods. A more extensive comparison shows that the two best performing algorithms, the MA and the SO, complement each other. In addition, the proposed method performs very well for cases where the number of facilities are low. If we consider only the cases where p≤ 50, we see that the MA performs best 5 times, 14 times and 14 times for α= 1, 2, 3, respectively, and loses twice, three times, and three times.

Exact solution approaches for the discrete α-neighbor p-center problem.

The discrete -neighbor -center problem (d--CP) is an emerging variant of the classical -center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate facilities on these points in such a way that the maximum distance between each point where no facility is located and its -closest facility is minimized. The only existing algorithms in literature for solving the d--CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d--CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&Calgorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.

The discrete [formula omitted]-center location problem with upgrading

In this paper, different upgrading strategies are investigated in the context of the p-center problem. The possibility of upgrading a set of connections to different centers is considered as well as the possibility of upgrading entire centers, i.e., all connections made to them. Two variants for these perspectives are analyzed: in the first, there is a limit on the number of connections or centers that can be upgraded; in the second, an existing budget is assumed for the same purpose. Different mixed-integer linear programming models are introduced for those problems as well as data-driven lower and upper bounds. In most cases, an optimal solution can be obtained within an acceptable computing time using an off-the-shelf solver. Nevertheless, this is not the case for one particular family of problems. This motivated the development of a math-heuristic seeking high-quality feasible solutions in that specific case. Extensive computational experiments are reported highlighting the relevance of upgrading connections or centers in the context of the p-center problem.

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.

A double-oracle, logic-based Benders decomposition approach to solve the [formula omitted]-adaptability problem

The K-adaptability problem is a special case of adaptive robust optimization with discrete recourse that aims to prepare K solutions under uncertainty, and select among them upon full knowledge of the realized scenario. We propose a novel approach to solve K-adaptability problems with linear objective and constraints, binary first-stage decision variables, second-stage objective uncertainty, and a polyhedral uncertainty set. A logic-based Benders decomposition is applied to handle the first-stage decisions in a master problem, thus the Benders subproblem becomes a min–max–min robust combinatorial optimization problem. To solve the subproblem, a double-oracle algorithm that iteratively generates adverse scenarios and recourse decisions and assigns scenarios to K-subsets of the decisions by solving p-center problems is devised. Extensions of the proposed approach to handle parameter uncertainty in both the first-stage objective and the second-stage constraints, and for integer first-stage decision variables and nonlinear functions, are also provided. We show that, under mild conditions, the proposed algorithm converges to an optimal solution and terminates in a finite number of iterations. Numerical results obtained from experiments on benchmark instances of the adaptive shortest path problem, the regular knapsack problem, the asset liability-management problem, and a generic K-adaptability problem demonstrate the performance advantage of the proposed approach when compared to state-of-the-art methods in the literature.

Auxiliary data structures and techniques to speed up solving of the [formula omitted]-next center problem: A VNS heuristic

In this paper we study the p-next center problem and propose an adequate solution approach. The p-next center problem aims to minimizing the maximum distance from a user to the nearest center plus the distance between the center and its closest center. In this paper we propose a new Variable Neighborhood Search based algorithm to solve the p-next center problem. It uses refined local search and shaking procedures as well as auxiliary data structures. The implementation consists in filtering out the candidate centers to enter a solution by considering only ones that potentially decrease the objective function value. The same approach has been applied to the classical p-center problem. Here we show that known properties of an efficient implementation of VNS heuristic developed for the p-center problem, hold for the new problem as well. More precisely, all the proposals in this work are inspired by other analogous ones used in the literature for similar problems. Hence, the novelty is the adaptation of the known properties that hold for the p-center problem to the p-next center problem. The performance of the proposed heuristic is assessed on the benchmark instances from the literature as well as newly generated larger instances with 1000, 1500, 2000 and 2500 vertices and instances defined over graphs up to 1000 vertices with different densities. The obtained results clearly demonstrate the effectiveness and efficiency of the proposed algorithm. Hence, the paper shows that the same observations used to solve p-center problem may be used to efficiently solve the p-next center problem.

The [formula omitted]-center problem under locational uncertainty of demand points

The p-center problem is finding the location of p facilities among a set of n demand points such that the maximum distance between any demand point and its nearest facility is minimized. In this paper, we study this problem in the context of uncertainty, that is, the location of the demand points may change in a region like a disk or a segment, or belong to a finite set of points. We introduce Max-p-center and Min-p-center problems which are the worst and the best possible solutions for the p-center problem under such locational uncertainty. We propose approximation and parameterized algorithms to solve these problems under the Euclidean metric. Further, we study the MinMax Regret 1-center problem under uncertainty and propose a linear-time algorithm to solve it under the Manhattan metric as well as an O(n4) time algorithm under the Euclidean metric.

Solving the p-second center problem with variable neighborhood search

The p-center problem is a well-known and highly studied problem pertaining to the identification of p of the potential n center locations in such a way as to minimize the maximum distance between the users and the closest center. As opposed to the p-center, the p-second center problem minimizes the maximum sum of the distances from the users to the closest and the second closest centers. In this paper, we propose a new Variable Neighborhood Search based algorithm for solving the p-second center problem. Its performance is assessed on the benchmark instances from the literature. Moreover, to further evaluate the algorithm?s performance, we generated larger instances with 1000, 1500, 2000, and 2500 nodes and instances defined over graphs up to 1000 nodes with different densities. The obtained results clearly demonstrate the effectiveness and efficiency of the proposed algorithm.

Optimal refuge chamber position in underground mines based on tree network

Safety is a critical concern in underground mining. Operations should be designed according to a vision of zero accident and fatality. A refuge chamber is a station where worker lives are maintained throughout an emergency until they are saved. Given that workers perform their duties in different areas of underground, the number of refuge chambers and their locations are of paramount importance as workers must be able to access them rapidly. Furthermore, the task of refuge chamber localization needs to consider the fact that the number of workers in a different area and the structural characteristics and stability conditions of these areas will not be identical. This paper aims to determine the optimal locations of the refuge chambers in the underground network considered as a tree. Initially, a formula quantifying the importance of each worksite is developed. Then, two main strategies are presented as feasible localization solutions: A recursive solution to the ‘Absolute Weighted P-centre Problem’ and a spiral approach to the ‘Total Covering problem’. The functionalities of the proposed approaches are demonstrated by case studies. Results show that the proposed methods can find viable locations for refuge chambers while fulfilling varied maximum safety distances of each working area.

Dynamically second-preferred [formula omitted]-center problem

This paper deals with the Dynamically Second-preferred p-center Problem (DSpP). In this problem, customers’ preferences and subsets of sites that each customer is willing to accept as service centers are taken into account. It is assumed that centers can fail and, thus, the decision maker is risk-averse and makes his decision taking into account not only the most favourite centers of the customers but also the worst case situation whenever they evaluate their preferred second opportunity. Specifically, the new problem aims at choosing at most p centers so that each demand point can visit at least two acceptable centers and the maximum sum of distances from any demand point to any of its preferred centers plus the distance from any of the preferred centers to any of the centers the user prefers once he is there is minimized. The problem is NP-hard as an extension of the p-next center problem. The paper presents three different mixed-integer linear programming formulations that are valid for the problem. Each formulation uses different space of variables giving rise to some strengthening using valid inequalities and variable fixing criteria that can be applied when valid upper bounds are available. Exact methods are limited so that a heuristic algorithm is also developed to provide good quality solution for large size instances. Finally, an extensive computational experience has been performed to assess the usefulness of the formulations to solve DSpP using standard MIP solvers.

Exploiting flat subspaces in local search for [formula omitted]-Center problem and two fault-tolerant variants

In this paper, local search algorithms are proposed for the p-Center, α-Neighbour p-Center and p-Next Center facility location problems. The α-Neighbour p-Center and p-Next Center problems may be viewed as two fault-tolerant variants of the p-Center problem. The algorithm proposed for p-Center outperforms the most recent state-of-the-art metaheuristic for this problem using standard datasets. The proposed algorithm for p-Next Center also outperforms an existing, more complex, state-of-the-art metaheuristic for this problem. The algorithm proposed for α-Neighbour p-Center is the first metaheuristic for this problem, to the best of the author's knowledge. The proposed algorithms share a common design, which is the integration of the first-improvement local search with strategies to exploit flat subspaces in the search space. The overall success of this design paradigm motivates further investigation about its properties and applications to similar NP-hard optimisation problems.

A Trade-Off Algorithm for Solving p-Center Problems with a Graph Convolutional Network

The spatial optimization method between combinatorial optimization problems and GIS has many geographical applications. The p-center problem is a classic NP-hard location modeling problem, which has essential applications in many real-world scenarios, such as urban facility locations (ambulances, fire stations, pipelines maintenance centers, police stations, etc.). This study implements two methods to solve this problem: an exact algorithm and an approximate algorithm. Exact algorithms can get the optimal solution to the problem, but they are inefficient and time-consuming. The approximate algorithm can give the sub-optimal solution of the problem in polynomial time, which has high efficiency, but the accuracy of the solution is closely related to the initialization center point. We propose a new paradigm that combines a graph convolution network and greedy algorithm to solve the p-center problem through direct training and realize that the efficiency is faster than the exact algorithm. The accuracy is superior to the heuristic algorithm. We generate a large amount of p-center problems by the Erdos–Renyi graph, which can generate instances in many real problems. Experiments show that our method can compromise between time and accuracy and affect the solution of p-center problems.

Efficient algorithms for the minmax regret path center problem with length constraint on trees

When there is no constraint on the length, efficient algorithms are known for the minmax regret path center, path median, and path centdian problems on trees. In a recent review on location problems, these problems with length constraint were considered as open. The focus of this paper is the minmax regret path center problem with length constraint on a tree. Efficient algorithms are presented for both the continuous and discrete models, which require, respectively, O(nlg2n) and O(nlgn) time. Our algorithms are based on an approach introduced by Averbakh and Berman for solving the minmax regret p-center problem. To apply their approach, we give a sufficient condition under which the approach works. This result is of independent interest. According to the condition, polynomial algorithms are obtained immediately for the minmax regret versions of many other center location problems, for which no algorithms were known before.

GRASP with strategic oscillation for the [formula omitted]-neighbor [formula omitted]-center problem

This paper presents a competitive algorithm that combines the Greedy Randomized Adaptive Search Procedure including a Tabu Search instead of a traditional Local Search framework, with a Strategic Oscillation post-processing, to provide high-quality solutions for the α-neighbor p-center problem (α−pCP). This problem seeks to locate p facilities to service or cover a set of n demand points with the objective of minimizing the maximum distance between each demand point and its αth nearest facility. The algorithm is compared to the best method found in the state of the art, which is an extremely efficient exact procedure for the continuous variant of the problem. An extensive comparison shows the relevance of the proposal, being able to provide competitive results independently of the α value.

A scaleable projection‐based branch‐and‐cut algorithm for the p‐center problem

The p-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose p of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds.We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation.Finally, we also present a computational study on instances from the literature with up to more than 700,000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters.

Two lower-bounding algorithms for the p-center problem in an area

The p-center location problem in an area is an important yet very difficult problem in location science. The objective is to determine the location of p hubs within a service area so that the distance from any point in the area to its nearest hub is as small as possible. While effective heuristic methods exist for finding good feasible solutions, research work that probes the lower bound of the problem’s objective value is still limited. This paper presents an iterative solution framework along with two optimization-based heuristics for computing and improving the lower bound, which is at the core of the problem’s difficulty. One method obtains the lower bound via solving the discrete version of the Euclidean p-center problem, and the other via solving a relatively easier clustering problem. Both methods have been validated in various test cases, and their performances can serve as a benchmark for future methodological improvements.