The Generalized Minimum Manhattan Network Problem (GMMN) – Scale-Diversity Aware Approximation and a Primal-Dual Algorithm

In the d-dimensional Generalized Minimum Manhattan Network (d-GMMN) problem one is interested in finding a minimum cost rectilinear network N connecting a given set R of n pairs of points in R such that each pair is connected in N via a shortest Manhattan path. The problem is known to be NP-complete and does not admit a FPTAS, the best known upper bound is an O(log n)-approximation for d > 2 and an O(log n)-approximation for d = 2 by Das et al. [3]. In this paper we provide some more insight into the problem and develop two new algorithms, a ‘scalediversity aware’ algorithm with an O(D) approximation guarantee for d = 2. Here D is a measure for the different ‘scales’ that appear in the input, D ∈ O(log n) but potentially much smaller depending on the problem instance. Moreover, this implies that a potential proof of O(1)-inapproximability for 2-GMMN requires gadgets of many different scales in the construction. The other algorithm is based on a primal-dual scheme solving a more general path covering problem. On 2-GMMN it performs pretty well in practice with good a posteriori, instance-based approximation guarantees. Furthermore, it can be extended naturally to deal with obstacle avoiding requirements.