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Register a new userBottleneck Non-Crossing Matching in the Plane
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Added by
Karim Abu-Affash
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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Let $P$ be a set of $2n$ points in the plane, and let $M_{rm C}$ (resp., $M_{rm NC}$) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of $P$. We study the problem of computing $M_{rm NC}$. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an $O(n^{1.5}log^{0.5} n)$-time algorithm that computes a non-crossing matching $M$ of $P$, such that $bn(M) le 2sqrt{10} cdot bn(M_{rm NC})$, where $bn(M)$ is the length of a longest edge in $M$. An interesting implication of our construction is that $bn(M_{rm NC})/bn(M_{rm C}) le 2sqrt{10}$.
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Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $pin S$, $qin T$ and for each $rin Scup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(nlog n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2cdot poly(log n))$ time exact algorithm and an $O( n^{3/2}cdot poly(log n))$ time $(1+epsilon)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].
Research about crossings is typically about minimization. In this paper, we consider emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.
We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing {Gamma} of G in the plane such that the edges of S are not crossed in {Gamma} by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.
Given a set $P$ of $n$ red and blue points in the plane, a emph{planar bichromatic spanning tree} of $P$ is a spanning tree of $P$, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Moreover, we present a polynomial-time $(8sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $lambda$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8sqrt{2}lambda$.
The geometric $delta$-minimum spanning tree problem ($delta$-MST) is the problem of finding a minimum spanning tree for a set of points in a normed vector space, such that no vertex in the tree has a degree which exceeds $delta$, and the sum of the lengths of the edges in the tree is minimum. The similarly defined geometric $delta$-minimum bottleneck spanning tree problem ($delta$-MBST), is the problem of finding a degree bounded spanning tree such that the length of the longest edge is minimum. For point sets that lie in the Euclidean plane, both of these problems have been shown to be NP-hard for certain specific values of $delta$. In this paper, we investigate the $delta$-MBST problem in $3$-dimensional Euclidean space and $3$-dimensional rectilinear space. We show that the problems are NP-hard for certain values of $delta$, and we provide inapproximability results for these cases. We also describe new approximation algorithms for solving these $3$-dimensional variants, and then analyse their worst-case performance.
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