Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. We consider the rectilinear instances of the TSP, i.e. TSP instances where the points lie in the plane and the distance between two points is the $l_1$ distance. We propose a fixed-parameter algorithm for the Rectilinear TSP running in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. Our approach can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$. It provides a $O\left(nh5^h\right)$ algorithm which improves significantly over the best known existing fixed-parameter algorithm for rectilinear Steiner tree.
['Hadrien Cambazard', 'Nicolas Catusse']
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane