Extremal Topological and Geometric Problems for Polyominoes

We give a complete solution to the extremal topological combinatorial problem of finding the minimum number of tiles needed to construct a polyomino with $h$ holes. We denote this number by $g(h)$ and we analyze structural properties of polyominoes with $h$ holes and $g(h)$ tiles, characterizing their efficiency by a topological isoperimetric inequality that relates minimum perimeter, the area of the holes, and the structure of the dual graph of a polyomino. For $h\leqslant 8$ the values of $g(h)$ were originally computed by Tomas Olivera e Silva in 2015, and for the sequence $h_l=(2^{2l}-1)/3$ by Kahle and Róldan-Roa in 2019, who also showed that asymptotically $g(h) \approx 2h$. Here we also prove that the sequence of polyominoes constructed by Kahle and Róldan-Roa that have $h_l=(2^{2l}-1)/3$ holes and $g(h_l)$ tiles, are in fact unique up to isometry with respect to attaining these extremal topological properties; that is, having the minimal number of tiles for $h_l$ holes.

THE TRAVELING SALESMAN PROBLEM FOR LINES AND RAYS IN THE PLANE

In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. In the path variant, we seek a shortest path that visits each region. We present several linear-time approximation algorithms with improved ratios for these problems for two cases of neighborhoods that are (infinite) lines, and respectively, (half-infinite) rays. Along the way we derive a tight bound on the minimum perimeter of a rectangle enclosing an open curve of length L.