Euclidean TSP with few inner points in linear space

TL;DR

This paper introduces a new linear space algorithm for the Euclidean Traveling Salesman Problem with few inner points, improving efficiency by combining divide-and-conquer with matching-based shrinking techniques.

Contribution

It extends planar separator methods with matching-based shrinking to achieve a linear space, faster algorithm for the Euclidean TSP with few inner points.

Findings

Achieves a linear space algorithm with $O(nk^2 + k^{O(\sqrt{k})})$ time complexity.

Demonstrates the problem admits a quadratic bikernel.

Extends divide-and-conquer methods with matching-based instance shrinking.

Abstract

Given a set of $n$ points in the Euclidean plane, such that just $k$ points are strictly inside the convex hull of the whole set, we want to find the shortest tour visiting every point. The fastest known algorithm for the version when $k$ is significantly smaller than $n$, i.e., when there are just few inner points, works in $O(k^{11\sqrt{k}} k^{1.5} n^{3})$ time [Knauer and Spillner, WG 2006], but also requires space of order $k^{c\sqrt{k}}n^{2}$. The best linear space algorithm takes $O(k! k n)$ time [Deineko, Hoffmann, Okamoto, Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a linear space $O(nk^2+k^{O(\sqrt{k})})$ time algorithm. The new insight is extending the known divide-and-conquer method based on planar separators with a matching-based argument to shrink the instance in every recursive call. This argument also shows that the problem admits a quadratic bikernel.