Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

TL;DR

This paper introduces fixed-parameter algorithms for rectilinear TSP and Steiner tree problems in the plane, achieving improved time complexities by exploiting the structure of points on horizontal lines.

Contribution

It presents novel fixed-parameter algorithms for rectilinear TSP and Steiner tree problems, improving upon previous time bounds by leveraging bounded-treewidth graph techniques.

Findings

Rectilinear TSP solved in O(nh7^h) time.

Rectilinear Steiner tree solved in O(nh5^h) time.

Both algorithms outperform previous best bounds.

Abstract

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. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.