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- Adamaszek, Anna, et al.
(författare)
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Approximation schemes for capacitated geometric network design
- 2018
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Ingår i: SIAM Journal on Discrete Mathematics. - 0895-4801. ; 32:4, s. 2720-2746
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Tidskriftsartikel (refereegranskat)abstract
- We study a capacitated network design problem in a geometric setting. The input consists of an integral edge capacity k and two sets of points on the Euclidean plane, sources, and sinks, with an integral demand for each point. The demand of each source specifies the amount of flow that has to be shipped from the source, and the demand of each sink specifies the amount of flow that has to be shipped to the sink. The goal is to construct a minimum-length network that allows one to route the requested flow from the sources to the sinks and where each edge in the network has capacity k. The vertices of the network are not constrained to the sets of sinks and sources-any point on the Euclidean plane can be used as a vertex. The flow is splittable and parallel edges are allowed. The capacitated geometric network design problem generalizes, among others, the geometric Steiner tree problem, and as such it is NP-hard. We show that if the demands are polynomially bounded and the edge capacity k is not too large, the single-sink capacitated geometric network design problem admits a polynomial time approximation scheme. If the capacity is arbitrarily large, then we design a quasi-polynomial time approximation scheme for the capacitated geometric network design problem allowing for an arbitrary number of sinks. Our results rely on a derivation of an upper bound on the number of vertices different from sources and sinks (the so-called Steiner vertices) in an optimal network. The bound is polynomial in the total demand of the sources.
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| 2. |
- Adamaszek, Anna, et al.
(författare)
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PTAS for k-tour cover poblem on the plane for moderately large values of k
- 2010
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Ingår i: International Journal of Foundations of Computer Science. - 0129-0541. ; 21:6, s. 893-904
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Tidskriftsartikel (refereegranskat)abstract
- Let P be a set of n points in the Euclidean plane and let O be the origin point in the plane. In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P, such that each tour starts and ends in O and covers at most k points from P. The k-tour cover problem is known to be NP-hard. It is also known to admit constant factor approximation algorithms for all values of k and even a polynomial-time approximation scheme (PTAS) for small values of k, k = circle divide (log n/ log log n). In this paper, we significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k <= 2(log delta n), where delta = delta(epsilon). The main technical result proved in the paper is a novel reduction of the k-tour cover problem with a set of n points to a small set of instances of the problem, each with circle divide((k/epsilon)(circle divide(1))) points.
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