Tiler

Tiler is a CLI tool for automated solving of polyomino tiling problems.

Currently, it features the following solver backends:

• simple backtracking (with a few tile selection heuristics)
• Donald Knuth's Algorithm X (Dancing Links)
• conversion to the SAT problem (several modifications)
• conversion to the ILP problem
• conversion to a MiniZinc constraint satisfaction problem

Usage

The CLI takes input (either from the command line or from a file) which defines a board (a region consisting of unit squares) and polyomino tiles and tries to find a perfect tiling of the board with the tiles.

The shape of the board as well as of each tile can be provided in any of these formats:

• name, eg. 3L (see the shapes command for the list of all named shapes)

• dimensions, eg. 2x4 (applicable only for rectangles)

• map, eg. this equivalent of 3x2:

  xxx
  xxx

  (useful with the -i option)

• perimeter, eg. DDRRULUL - an equivalent of 3L (crossings of the perimeter are not allowed, applicable only for shapes without holes)

You should provide a list of shapes, where the first one represents the board and the others the tiles. Tile shape may be prefixed with N: where N is the available number of such tiles (unlimited if not provided). See the example command for input examples.

Run tiler -h and tiler solve -h to see all options.

Limitations

Only continuous shapes are allowed.

The solver may rotate the tiles. There is a CLI flag to allow reflections (flipping the tiles over).

Examples

Here are some example outputs of the solver.

tiler solve -o 01.svg 11x7 4T 3:3I

tiler solve -o 02.svg 10x9 1:5A 1:5F 1:5G 1:5J 1:5P 1:5S 1:5Y 1:5R 1:5N 1:5L 1:5Q 1:5Z 1:5I 1:5T 1:5U 1:5V 1:5W 1:5X

Build Instructions

Before building and running Tiler, always run source scripts/load-env.sh to load the environment variables required by some dependencies.

Dependencies

There are several optional open-source dependencies (in the external directory):

• DLX1 exact cover finding algorithm by Donald Knuth
• CaDiCaL SAT solver
• CryptoMiniSat SAT solver
• Kissat SAT solver
• BreakID CNF symmetry breaking library
• SBVA CNF preprocessor
• PBLib pseudo-boolean constraint encoding library (required when either CaDiCaL, CryptoMiniSat or Kissat is present)
• MiniZinc constraint modeling framework
• Gecode constraint solver (required when MiniZinc is present)
• Chuffed constraint solver (optional when MiniZinc is present)

Run ./scripts/build-all-deps.sh to clone and build them. You can select only some of them by running a subset of ./scripts/build-*.sh scripts and setting CMake options -DDLX=OFF / -DCADICAL=OFF / -DCRYPTOMINISAT=OFF / -DKISSAT=OFF / -DBREAKID=OFF / -DSBVA=OFF / -DMINIZINC=OFF / -DCHUFFED=OFF. Having the libraries installed system-wide should work as well (at least for some of them).

Additionally, Tiler can be dynamically linked against the Gurobi (v10.0.2) ILP solver. If you have an appropriate license, follow the official Gurobi installation guide. Otherwise, set the CMake option -DGUROBI=OFF.

Building Tiler

Building Tiler requires these dependencies:

• CMake build tool (version 3.12 or later)
• Conan package manager
• a C/C++ compiler (a recent version of GCC is recommended, C++17 support is required)

Build the executable by running ./scripts/release-setup.sh and ./scripts/release-build.sh. You can configure build options using eg. ccmake before running release-build.sh or by adding appropriate -D flags to release-setup.sh (they will be passed on to CMake).

Your freshly compiled Tiler executable will be located at release/bin/tiler.

This setup is tested on Linux with GCC 10.2.0, but it should work on other platforms, too (maybe with some minor tweaks).

Development Setup

In addition to CMake and Conan (see Build Instructions), you should install:

• ClangFormat code formatter (version 10 or later)
• Cppcheck static code analysis tool
• GDB debugger (optional)
• Python 3, Pandas and Matplotlib (if you want to run the benchmark displaying script)

Regarding the actual development, using VS Code with the C/C++ extension is recommended - this way, you can use the IDE tasks, the built-in debugging UI and utilize the proposed VS Code settings. If you intend to use a different editor/IDE, all commands to run can be found at tasks.json.

Benchmarks

To run the benchmarking suite on all problems in the problems directory, run ./scripts/release-benchmark.sh (however, note that this will take several hours; you can edit the benchmark_filter parameter in the script to select only some problems or solvers). You can use ./benchmark/plot.py to plot the measured data.

Problems

The problems directory contains several tiling problems for testing and benchmarking. They are organized by their tile sets, and their filenames try to describe the respective problems (e.g. they usually contain board dimensions as a measure of the problem size). Each filename ends with either _s, if it is solvable, or _u, if it is unsolvable. If the problem is intended to be solved with reflections allowed, its filename contains a quote '.

The following books were used as sources of some problems:

S. W. Golomb. Polyominoes: Puzzles, patterns, problems, and packings. Princeton University Press, 2nd edition, 1994.

many-unique/4r-5r-04x20'_s
many-unique/4r-5r-05x16'_s
many-unique/4r-5r-08x10'_s
many-unique/5r-03x20'_s
many-unique/5r-04x15'_s
many-unique/5r-05x12'_s
many-unique/5r-06x10'_s
many-unique/5r-diamond'_u
many-unique/5r-fcr'_s
many-unique/5r-H-08x09'_s
many-unique/5r-hskew-06x10'_u
many-unique/5r-plus'_u
many-unique/5r-sideways'_u
many-unique/5r-skew-03x20'_s
many-unique/5r-skew-04x15'_s
many-unique/5r-skew-05x12'_s
many-unique/5r-skew-06x10'_s

G. E. Martin. Polyominoes: a Guide to Puzzles and Problems in Tiling. Mathematical Association of America, 1991.

4I/10x10_u
4L/10x10_u
mixed/1-3I-07x07_s
mixed/4Os-4T-06x06_u

The remaining problems are either modifications of the listed ones, trivial, generally-known, or genuine.

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License

Tiler is licensed under the GNU GPL license v3. This does not apply to files in the external directory.

Tiler - tool for automated solving of polyomino tiling problems
Copyright (C) 2020  David Misiak

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.