Bottleneck Non-crossing Matching in the Plane

Let P be a set of 2n points in the plane, and letMC (resp.,MNC) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of P . We study the problem of computing MNC. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an O(n log n)-time algorithm that computes a noncrossing matching M of P , such that bn(M) ≤ 2 √ 10 · bn(MNC), where bn(M) is the length of a longest edge in M . An interesting implication of our construction is that bn(MNC)/bn(MC) ≤ 2 √ 10. ∗Work by A.K. Abu-Affash was partially supported by a fellowship for doctoral students from the Planning & Budgeting Committee of the Israel Council for Higher Education, and by a scholarship for advanced studies from the Israel Ministry of Science and Technology. Work by A.K. Abu-Affash and Y. Trabelsi was partially supported by the Lynn and William Frankel Center for Computer Sciences. Work by P. Carmi was partially supported by grant 2240-2100.6/2009 from the German Israeli Foundation for scientific research and development, and by grant 87212211 from the Israel Science Foundation. Work by M. Katz was partially supported by grant 1045/10 from the Israel Science Foundation. Work by M. Katz and P. Carmi was partially supported by grant 2010074 from the United States – Israel Binational Science Foundation.