On Covering Segments with Unit Intervals

We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem. We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration. We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise

Приближенные алгоритмы с гарантированными оценками точности для пересечения множеств ребер некоторых метрических графов равными кругами

Polynomial-time approximation algorithms with constant approximation ratio are proposed for the problem of intersection of a given set of n planar straight line segments with the least number of equal disks. In the case where the segments have at most k different orientations, a simple 4k-approximate algorithm with time complexity O(n log n) is known. In addition, a 100-approximate algorithm with time complexity O(n4 log n) is known for the case of the problem on the edge sets of plane graphs. In this paper, for instances of the problem on the edge sets of Gabriel graphs, relative neighbourhood graphs, and Euclidean minimum spanning trees, in which the number of different edge orientations is, in general, unbounded, we construct simple O(n2)-time approximation algorithms with approximation ratios 14, 12, and 10, respectively. These algorithms outperform the aforementioned approximation algorithm for the general setting of the problem for edge sets of plane graphs. © 2019 Krasovskii Institute of Mathematics and Mechanics. All rights reserved