Bottleneck Non-crossing Matching in the Plane

Approximating the bottleneck plane perfect matching of a point set

“…Abu-Affash et al [1] showed that the bottleneck plane perfect matching problem is NP-hard and presented an algorithm that computes a plane perfect matching whose edges have length at most 2 √ 10 times the bottleneck, i.e., 2 √ 10λ * . They also showed that this problem does not admit a PTAS (Polynomial Time Approximation Scheme), unless P=NP.…”

“…On the other hand the weight of the bottleneck matching can be unbounded with respect to the weight of the minimum weight matching, see Figure 1. Matching and bottleneck matching problems play an important role in graph theory, and thus, they have been studied extensively, e.g., [1,2,4,7,8,11,12]. Self-crossing configurations Table 1: Summary of results.…”

“…The bottleneck, λ * , is the length of the longest edge in M * . The problem of computing M * has been proved to be NP-hard [1]. Figure 1 in [1] and [3] shows that the longest edge in the minimum weight matching (which is planar) can be unbounded with respect to λ * .…”

“…The problem of computing M * has been proved to be NP-hard [1]. Figure 1 in [1] and [3] shows that the longest edge in the minimum weight matching (which is planar) can be unbounded with respect to λ * . On the other hand the weight of the bottleneck matching can be unbounded with respect to the weight of the minimum weight matching, see Figure 1.…”

“…Self-crossing configurations Table 1: Summary of results. Algorithm time complexity bottleneck (λ) size of matching Abu-Affash et al [1] O(n 1.5 √ log n) 2 √ 10λ * n/2 Section 4.1 O(n log 2 n) λ * n/5 Section 4.2 O(n log n) (…”