Tile Packing Tomography is NP-hard

TL;DR

This paper proves that reconstructing tile packings from row and column projections is NP-hard for all tiles that are not bars, extending the understanding of the computational complexity of discrete tomography problems.

Contribution

It establishes NP-hardness for tile packing reconstruction for all non-bar tiles, generalizing previous results and highlighting the problem's computational difficulty.

Findings

NP-hardness for non-bar tiles in tile packing reconstruction

Greedy algorithm solves bar tile cases efficiently

Complexity extends to all non-bar tile shapes

Abstract

Discrete tomography deals with reconstructing finite spatial objects from lower dimensional projections and has applications for example in timetable design. In this paper we consider the problem of reconstructing a tile packing from its row and column projections. It consists of disjoint copies of a fixed tile, all contained in some rectangular grid. The projections tell how many cells are covered by a tile in each row and column. How difficult is it to construct a tile packing satisfying given projections? It was known to be solvable by a greedy algorithm for bars (tiles of width or height 1), and NP-hardness results were known for some specific tiles. This paper shows that the problem is NP-hard whenever the tile is not a bar.