New Results in Sona Drawing: Hardness and TSP Separation

TL;DR

This paper proves that finding the shortest sona drawing for a set of points is NP-hard and can be significantly longer than the TSP tour, also establishing NP-hardness for grid-based tours, answering longstanding open questions.

Contribution

It establishes NP-hardness results for sona drawing length minimization and grid-based TSP tours, providing new complexity insights and answering questions from CCCG 2006.

Findings

Minimum-length sona drawing is NP-hard.

Sona drawing can be longer than TSP tour by a factor > 1.5487875.

Deciding existence of grid-based tours is NP-hard.

Abstract

Given a set of point sites, a sona drawing is a single closed curve, disjoint from the sites and intersecting itself only in simple crossings, so that each bounded region of its complement contains exactly one of the sites. We prove that it is NP-hard to find a minimum-length sona drawing for $n$ given points, and that such a curve can be longer than the TSP tour of the same points by a factor $> 1.5487875$. When restricted to tours that lie on the edges of a square grid, with points in the grid cells, we prove that it is NP-hard even to decide whether such a tour exists. These results answer questions posed at CCCG 2006.