Long Non-crossing Configurations in the Plane

Abstract

We revisit some maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of n points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching, (b) a Hamiltonian path, and (c) a spanning tree. We obtain some new results for (b) and (c), as well as for the Hamiltonian cycle problem:(i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio 2 π+1 ≈ 0.4829. The previous best ratio, due to Alon et al., was 1 π ≈ 0.3183. The ratio of our algorithm is close to 2 π ≈ 0.6366 on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. For instance "random" point sets meet the condition with high probability. The algorithm runs in O(n 7/3 log n) time.(ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio 0.502 which runs in O(n log n) time. The previous ratio, 1/2, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster algorithm and a very simple analysis.(iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to 2/π on a relatively broad class of instances: for point sets where the product diameter × convex hull size is much smaller than the maximum length matching. Again "random" point sets meet the condition with high probability. However, this algorithm does not come with a constant approximation guarantee for all instances. The algorithm runs in O(n 7/3 log n) time. No previous approximation results were known for this problem.

Bounds on the Maximum Multiplicity of Some Common Geometric Graphs

“…Nothing is known about the complexity of computing the longest non-crossing spanning tour, spanning path, perfect matching or spanning tree. However, constant ratio approximations are available for the last three problems; see [3,12].…”

“…Note that the simple polygon bounded by an almost convex chain P + (n, k r ) and the segment between its two extremal vertices has Θ * (t(n, k r )) triangulations. ) with opposite orientations that form D(38, 2 12 ). Two consecutive hull vertices of P + (19, 2 6 ) with a reflex chain of two vertices in between are indicated in both the upper and the lower chain.…”

On the Longest Spanning Tree with Neighborhoods

“…While past research has primarily focused on minimization problems, the maximization variants usually require different techniques and so they are interesting in their own right and pose many unmet challenges; e.g., see the section devoted to longest subgraph problems in the survey of Bern and Eppstein [5]. The results obtained in this area in the last 20 years are rather sparse; the few articles [4,8,10] make a representative sample.…”

“…Using a technique developed in [8] (in fact a simplification of an earlier approach used in [2]), we first obtain a simple approximation algorithm with ratio 1/2. Algorithm A1.…”

On the Heaviest Increasing or Decreasing Subsequence of a Permutation, and Paths and Matchings on Weighted Point Sets

“…The problem of finding non-crossing paths, cycles and trees with vertices on point sets on the plane optimizing some given functions, has been of interest to many computational geometers for some time now. Alon, Rajagopalan, and Suri [1], and Dumitrescu and Tóth [3] studied the problem of finding non-crossing paths, matchings, cycles, and trees of maximum length, where the length od a cycle, matching, and tree, is the sum of the lengths of its edges.…”