Computational geometry: an introduction
Computational geometry: an introduction
On a multidimensional search technique and its application to the Euclidean one centre problem
SIAM Journal on Computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
Discrete & Computational Geometry
Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Algebraic optimization: the Fermat-Weber location problem
Mathematical Programming: Series A and B
Handbook of discrete and computational geometry
Efficient algorithms for geometric optimization
ACM Computing Surveys (CSUR)
Mathematics of Operations Research
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
On the continuous Weber and k-median problems (extended abstract)
Proceedings of the sixteenth annual symposium on Computational geometry
Minmax Regret Median Location on a Network Under Uncertainty
INFORMS Journal on Computing
Complexity of robust single facility location problems on networks with uncertain edge lengths
Discrete Applied Mathematics
Minmax regret solutions for minimax optimization problems with uncertainty
Operations Research Letters
Improved algorithms for the minmax-regret 1-center and 1-median problems
ACM Transactions on Algorithms (TALG)
Minimax Regret Single-Facility Ordered Median Location Problems on Networks
INFORMS Journal on Computing
Largest bounding box, smallest diameter, and related problems on imprecise points
Computational Geometry: Theory and Applications
Locating emergency facilities with random demand for risk minimization
Expert Systems with Applications: An International Journal
Largest bounding box, smallest diameter, and related problems on imprecise points
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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We consider single facility location problems (1-median and weighted 1-center) on a plane with uncertain weights and coordinates of customers (demand points). Specifically, for each customer, only interval estimates for its weight and coordinates are known. It is required to find a ''minmax regret'' location, i.e. to minimize the worst-case loss in the objective function value that may occur because the decision is made without knowing the exact values of customers' weights and coordinates that will get realized. We present an O(n^2log^2n) algorithm for the interval data minmax regret rectilinear 1-median problem and an O(nlogn) algorithm for the interval data minmax regret rectilinear weighted 1-center problem. For the case of Euclidean distances, we consider uncertainty only in customers' weights. We discuss possibilities of solving approximately the minmax regret Euclidean 1-median problem, and present an O(n^22^@a^(^n^)log^2n) algorithm for solving the minmax regret Euclidean weighted 1-center problem, where @a(n) is the inverse Ackermann function.