The Transitive Minimum Manhattan Subnetwork Problem in 3 dimensions

作者：Engels Birgit^*

来源：Discrete Applied Mathematics, 2010, 158(4): 298-307.

DOI：10.1016/j.dam.2009.06.025

• Manhattan network
• 1-spanner
• Grid graph
• 3 dimensions

We consider the Minimum Manhattan Subnetwork (MMSN) Problem which generalizes the already known Minimum Manhattan Network (MMN) Problem: Given a set P of n points in the plane, find shortest rectilinear paths between all pairs of points. These paths form a network, the total length of which has to be minimized. From a graph theoretical point of view, a MMN is a 1-spanner with respect to the L(1) metric. In contrast to the MMN problem, a solution to the MMSN problem does not demand L(1)-shortest paths for all point pairs, but only for a given set R subset of P x P of pairs. The complexity status of the MMN problem is still unsolved in >= 2 dimensions, whereas the MMSN was shown to be NP-complete considering general relations R in the plane. We restrict the MMSN problem to transitive relations R(T) (Transitive Minimum Manhattan Subnetwork (TMMSN) Problem) and show that the TMMSN problem in 3 dimensions is NP-complete.

• 出版日期2010-2-28