Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
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)
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 n points P@?R^d and functions @a:P-[0,~) and @b:P-[1,~). We wish to compute a subset T@?P and a salesman tour @p 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 @a(p)+@b(p)@?d(p,q), where q@?T is the nearest point on the tour. We give a PTAS with complexity O(nlog^d^+^3n) for this problem. Moreover, we propose an O(npolylog(n))-time PTAS for the Steiner variant of this problem. This dramatically improves a recent result of Armon et al. [A. Armon, A. Avidor, O. Schwartz, Cooperative TSP, in: Proceedings of the 14th Annual European Symposium on Algorithms, 2006, pp. 40-51].