Dynamic Programming Approach to the Generalized Minimum Manhattan Network Problem

We study the generalized minimum Manhattan network (GMMN) problem: given a set $$P$$ of pairs points in Euclidean plane $${\mathbb{R}}^2$$ , we are required to find minimum-length geometric which consists axis-aligned segments and contains shortest path $$L_1$$ metric (a so-called path) for each pair . This problem commonly generalizes several NP-hard design problems that admit constant-factor approximation algorithms, such as rectilinear Steiner arborescence (RSA) problem, it is open whether so does GMMN problem. As bottom-up exploration, Schnizler (2015) focused on intersection graphs rectangles defined by gave polynomial-time dynamic programming algorithm whose input restricted both treewidth maximum degree its graph bounded constants. In this paper, first attempt remove bound, provide star case, extend general tree case based an improved approach.