Approximation Algorithms for Piercing Special Families of Hippodromes: An Extended Abstract

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

“…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.…”

“…In [13] faster O(n 2 )-time 10-, 12-and 14-approximate algorithms are designed for the NP-hard [10] IPGD problem when E is being an edge set of a minimum Euclidean spanning tree, a relative neighborhood graph and a Gabriel graph respectively. This paper extends this latter work by presenting faster O(n 3/2 log 2 n)-expected time 10-, 12-and 14-approximation algorithms for the IPGD problem for classes of minimum Euclidean spanning trees, relative neighborhood graphs and Gabriel graphs respectively.…”

“…Our subquadratic O(1)-approximation algorithms are improved versions of the O(n 2 )-time O(1)-approximation algorithms, given in [13]. The latter algorithms operate on two concepts whose definitions are given below.…”

“…The latter algorithms operate on two concepts whose definitions are given below. It turns out (see Lemmas 1 and 5 in [13] for proof) that any edge of every Gabriel and relative neighborhood graph is α-coverable for some suitable positive integer α.…”

“…It can also be proved (see the lemma 3 from [13]) that in any subgraph of a minimum Euclidean spanning tree there always exists a 10-coverable edge. Lemma 3.…”