Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: {A} Factor 2.5 Approximation Algorithm
Résumé
Let ${\mathcal B}$ be a centrally symmetric convex polygon of ${\mathbb R}^2$ and $||{\bf p}-{\bf q}||$ be the distance between
two points ${\bf p},{\bf q}\in {\mathbb R}^2$ in the normed plane whose unit ball is ${\mathcal B}$. For a set $T$ of $n$ points (terminals) in ${\mathbb R}^2$, a ${\mathcal B}$-network on $T$ is a network $N(T)=(V,E)$ with the property that its edges are parallel to the directions of ${\mathcal B}$ and for every pair of terminals ${\bf t}_i$ and ${\bf t}_j$, the network $N(T)$ contains a shortest ${\mathcal B}$-path between them, i.e., a path of length $||{\bf t}_i-{\bf t}_j||.$ A minimum ${\mathcal B}$-network on $T$ is a ${\mathcal B}$-network of minimum possible length. The problem of finding minimum ${\mathcal B}$-networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case
when the unit ball ${\mathcal B}$ is a square (and hence the distance $||{\bf p}-{\bf q}||$ is the $l_1$ or the $l_{\infty}$-distance between ${\bf p}$ and ${\bf q}$) and it has been shown recently by Chin, Guo, and Sun \cite{ChGuSu} to be strongly NP-complete. Several approximation algorithms (with factors 8,4,3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum ${\mathcal B}$-network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper \cite{ChNouVa} and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.