Approximation schemes for Euclidean k-medians and related problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Approximating geometrical graphs via “spanners” and “banyans”
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
Improved Greedy Algorithms for Constructing Sparse Geometric Spanners
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
A Nearly Linear-Time Approximation Scheme for the Euclidean kappa-median Problem
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
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The (Euclidean) Vehicle Routing Allocation Problem (VRAP) is a generalization of Euclidean TSP. We do not require that all points lie on the salesman tour. However, points that do not lie on the tour are allocated, i.e., they are directly connected to the nearest tour point, paying a higher (per-unit) cost. More formally, the input is a set of points P ⊂ Rd and functions α : P → [0,∞) and ß : P → [1,∞). We wish to compute a subset T ⊆ P and a salesman tour π through T such that the total length of the tour plus the total allocation cost is minimum. The allocation cost for a single point p ∈ P\T is α(p) + ß(p) ċ d(p, q), where q ∈ T is the nearest point on the tour. We give a PTAS with complexity O(n logd+3 n) for this problem. Moreover, we propose a O(n polylog (n))-time PTAS for the Steiner variant of this problem. This dramatically improves a recent result of Armon et al. [2].