A Note on Tiling under Tomographic Constraints

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.Comment: added one author and a few theorem

Tile Packing Tomography is NP-hard

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

The reconstruction of a subclass of domino tilings from two projections

AbstractWe present a new way of studying the classical and still unsolved problem of the reconstruction of a domino tiling from its row and column projections. After giving a simple greedy strategy for solving the problem from one projection, we introduce the concept of degree of a domino tiling. We generalize an algorithm for the reconstruction of domino tilings of degree two from two projections, to domino tilings of degree three and four

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

On tiling under tomographic constraints

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 1×1 or 1×2 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