A Note on Tiling under Tomographic Constraints

TL;DR

This paper investigates the computational complexity of reconstructing 2D grid tilings from projection data, establishing NP-completeness for several tile sets to advance understanding of tiling reconstruction problems.

Contribution

It provides new NP-completeness results for various tiling reconstruction problems, moving toward a comprehensive classification of their computational complexity.

Findings

NP-completeness for certain tile sets in tiling reconstruction

Progress toward classifying complexity of tiling problems

Extension of previous polynomial-time and NP-complete results

Abstract

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling which has given projections. Some simple variants of this problem, involving tiles that are 1x1 or 1x2 rectangles, have been studied in the past, and were proved to be either solvable in polynomial time or NP-complete. In this note we make progress toward a comprehensive classification of various tiling reconstruction problems, by proving NP-completeness results for several sets of tiles.