Practical Approximation Algorithms for Stabbing Special Families of Line Segments with Equal Disks

Kobylkin, K. S.; Dryakhlova, Irina

doi:10.1007/978-3-030-53552-0_7

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“…Namely, a 100-approximate O(n 4 log n)-time algorithm is given for the problem in its general setting where E is formed by an edge set of an arbitrary plane graph. Moreover, due to applications, 68-and 54-approximate algorithms are given in [12] for special cases where E is an edge set of a generalized outerplane graph and a Delaunay triangulation respectively as well as a 23-approximation algorithm is proposed under the assumption that all pairs of non-overlapping segments from E are at the distance more than r from each other.…”

Section: Related Work and Our Resultsmentioning

confidence: 99%

Fast Approximation Algorithms for Stabbing Special Families of Line Segments with Equal Disks

Kobylkin

2021

Lecture Notes in Computer Science

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An NP-hard problem is considered to stab a given set of n straight line segments on the plane with the smallest size subset of disks of fixed radii r > 0, where the set of segments forms a straight line drawing G = (V, E) of a planar graph without proper edge crossings. To the best of our knowledge, only 100-approximation O(n 4 log n)-time algorithm is known (Kobylkin, 2018) for this problem. Moreover, when segments of E are axis-parallel, 8-approximation is proposed (Dash et al., 2012), working in O(n log n) time. In this work another special setting is considered of the problem where G belongs to classes of special plane graphs, which are of interest in network applications. Namely, three fast O(n 3/2 log 2 n)-expected time algorithms are proposed: a 10approximate algorithm for the problem, considered on edge sets of minimum Euclidean spanning trees, a 12-approximate algorithm for edge sets of relative neighborhood graphs and 14-approximate algorithm for edge sets of Gabriel graphs. The paper extends recent work (Kobylkin et al. 2019) where O(n 2 )-time approximation algorithms are proposed with the same constant approximation factors for the problem on those three classes of sets of segments.

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Section: Related Work and Our Resultsmentioning

confidence: 99%