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ON THE COMPLEXITY OF SOME COMMON GEOMETRIC LOCATION PROBLEMS
 SIAM J. COMPUTING
, 1984
"... Given n demand points in the plane, the pcenter problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo & point to its respective nearest supply point. The pmedian problem is to minimize the sum of distances from demand points to their resp ..."
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Cited by 118 (1 self)
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Given n demand points in the plane, the pcenter problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo & point to its respective nearest supply point. The pmedian problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the pcenter and the pmedia problems relative to both the Euclidean and the rectilinear metrics are NPhard. In fact, we prove that it is NPhard even to approximate the pcenter problems sufficiently closely. The reductions are from 3satisfiability.
The Quadratic Assignment Problem: A Survey and Recent Developments
 In Proceedings of the DIMACS Workshop on Quadratic Assignment Problems, volume 16 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1994
"... . Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment probl ..."
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Cited by 93 (16 self)
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. Quadratic Assignment Problems model many applications in diverse areas such as operations research, parallel and distributed computing, and combinatorial data analysis. In this paper we survey some of the most important techniques, applications, and methods regarding the quadratic assignment problem. We focus our attention on recent developments. 1. Introduction Given a set N = f1; 2; : : : ; ng and n \Theta n matrices F = (f ij ) and D = (d kl ), the quadratic assignment problem (QAP) can be stated as follows: min p2\Pi N n X i=1 n X j=1 f ij d p(i)p(j) + n X i=1 c ip(i) ; where \Pi N is the set of all permutations of N . One of the major applications of the QAP is in location theory where the matrix F = (f ij ) is the flow matrix, i.e. f ij is the flow of materials from facility i to facility j, and D = (d kl ) is the distance matrix, i.e. d kl represents the distance from location k to location l [62, 67, 137]. The cost of simultaneously assigning facility i to locat...
Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming
 Operations Research
, 1995
"... A typical example of the quadratic assignment problem (QAP) is the facility location problem, in which a set of n facilities are to be assigned, at minimum cost, to an equal number of locations. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the produc ..."
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Cited by 32 (3 self)
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A typical example of the quadratic assignment problem (QAP) is the facility location problem, in which a set of n facilities are to be assigned, at minimum cost, to an equal number of locations. Between each pair of facilities, there is a given amount of flow, contributing a cost equal to the product of the flow and the distance between locations to which the facilities are assigned. Proving optimality of solutions to quadratic assignment problems has been limited to instances of small dimension (n less than or equal to 20), in part because known lower bounds for the QAP are of poor quality. In this paper, we compute lower bounds for a wide range of quadratic assignment problems using a linear programmingbased lower bound studied by Drezner (1994). On the majority of quadratic assignment problems tested, the computed lower bound is the new best known lower bound. In 87 percent of the instances, we produced the best known lower bound. On several instances, including some of dimension n equal to 20, the lower bound is tight. The linear programs, which can be large even for moderate values of n, are solved with an interior point code that uses a preconditioned conjugate gradient algorithm to compute the directions taken at each iteration by the interior point algorithm. Attempts to
Rectilinear and Polygonal pPiercing and pCenter Problems
 In Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axispa ..."
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Cited by 28 (1 self)
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We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axisparallel rectangles or convex coriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions) . We also investigate the planar rectilinear (and polygonal) pcenter problem, in which we are given a set S of n points in the plane, and wish to find p axisparallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a lineartime solution for the rectilinear 3center problem (by showing that this problem can be formulated as an LPtype problem and by exhibiting a relation to Helly numbers). We give O(n log n...
Applications of Network Optimization
 Network Models, volume 7 of Handbooks in Operations Research and Management Science
, 1995
"... Network optimization has always been a core problem domain in operations research, as well as in computer science, applied mathematics, and many fields of engineering and management. Network optimization problems arise in a variety of situations, and often in situations that apparently are quite unr ..."
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Cited by 14 (0 self)
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Network optimization has always been a core problem domain in operations research, as well as in computer science, applied mathematics, and many fields of engineering and management. Network optimization problems arise in a variety of situations, and often in situations that apparently are quite unrelated to networks. These applications are scattered throughout the literature and until recently no single paper, book, or any other reference, summarized these applications. Consequently, the research and practitioner community has not fully appreciated the richness of these applications. This paper attempts to partially satisfy this important need by presenting a collection of applications of the following fundamental network optimization problems: the shortest path problem, the maximum flow problem, the minimum cost flow problem, assignment and matching problems, and the minimum spanning tree problem. We describe 25 applications of these problems and provide references for more than 100 additional applications. This paper is intended to provide an appreciation for the pervasiveness of network optimization problems. We hope that this paper will stimulate researchers and practitioners to model more decisions problems within the framework of network optimization.
Competitive Facility Location: The Voronoi Game
, 2003
"... We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearestneighbor rule, and the player whose points control the larger a ..."
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Cited by 11 (1 self)
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We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearestneighbor rule, and the player whose points control the larger area wins. We present a winning strategy for the second player, where the arena is a circle or a line segment. We also consider a variation where players can play more than one point at a time for the circle arena.
A Branch and Bound Algorithm for the Quadratic Assignment Problem using a Lower Bound Based on Linear Programming
 In C. Floudas and P.M. Pardalos, editors, State of the Art in Global Optimization: Computational Methods and Applications
, 1995
"... In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. Computational experience with the branch and bound algorithm on several Q ..."
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Cited by 9 (2 self)
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In this paper, we study a branch and bound algorithm for the quadratic assignment problem (QAP) that uses a lower bound based on the linear programming (LP) relaxation of a classical integer programming formulation of the QAP. Computational experience with the branch and bound algorithm on several QAP test problems is reported. The linear programming relaxations are solved with an implementation of an interior point algorithm that uses a preconditioned conjugate gradient algorithm to compute directions. The branch and bound algorithm is compared with a similar branch and bound algorithm that uses the GilmoreLawler lower bound (GLB) instead of the LPbased bound. The LPbased algorithm examines a small portion of the nodes explored by the GLBbased algorithm. 1 Introduction The quadratic assignment problem (QAP), first proposed by Koopmans and Beckmann [16], can be stated as min p2\Pi n X i=1 n X j=1 a ij b p(i)p(j) ; To appear in Proceedings of State of the Art in Global Opti...
Competitive Facility Location along a Highway
 In 7th Annual International Computing and Combinatorics Conference, volume 2108 of LNCS
, 2001
"... We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearestneighbor rule, and the player whose points control the larger a ..."
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Cited by 8 (3 self)
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We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearestneighbor rule, and the player whose points control the larger area wins. We present a winning strategy for the second player, where the arena is a circle or a line segment. We also consider a variation where players can play more than one point at a time for the circle arena.
A Simple Linear Time Algorithm for Proper Box Rectangular Drawing of Plane Graphs
 Journal of Algorithms
, 2000
"... In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is dra ..."
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Cited by 6 (0 self)
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In this paper we introduce a new drawing style of a plane graph G, called proper box rectangular (PBR ) drawing. It is defined to be a drawing of G such that every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal or a vertical line segment, and each face is drawn as a rectangle. We establish necessary and sufficient conditions for G to have a PBR drawing. We also give a simple linear time algorithm for finding such drawings. The PBR drawing is closely related to the box rectangular (BR ) drawing defined by Rahman, Nakano and Nishizeki [17]. Our method can be adapted to provide a new simpler algorithm for solving the BR drawing problem. 1 Introduction The problem of "nicely" drawing a graph G has received increasing attention [5]. Typically, we want to draw the edges and the vertices of G on the plane so that certain aesthetic quality conditions and/or optimization measures are met. Such drawings are very useful in visualizing planar graphs and fi...
Some constrained minimax and maximin location problems
 Studies in Locational Analysis
, 2000
"... In this paper we consider constrained versions of the Euclidean minimax facility location problem. We provide an O(n+m) time algorithm for the problem of constructing the minimum enclosing circle of a set of n points with center constrained to satisfy m linear constraints. As a corollary, we obtain ..."
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Cited by 5 (2 self)
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In this paper we consider constrained versions of the Euclidean minimax facility location problem. We provide an O(n+m) time algorithm for the problem of constructing the minimum enclosing circle of a set of n points with center constrained to satisfy m linear constraints. As a corollary, we obtain a linear time algorithm for the problem when the center is constrained to lie in an mvertex convex polygon, which improves the best known solution of O((n + m) log(n + m)) time. We also consider some constrained versions of the maximin problem, namely an obnoxious facility location problem in which we are given a set of n linear constraints, each representing a halfplane where some population may live, and the goal is to locate a point such that the minimum distance to the inhabited region is maximized. We provide optimal (n) time algorithms for this problem in the plane, as well as on the sphere.