1. Spiral Galaxies Puzzles are NP-complete
Erich Friedman
Stetson University
October 2, 2002

2. Spiral Galaxies Puzzles
puzzles consist of grid of squares and some circles.
the object is to divide a puzzle into connected groups of squares that contain one circle, which must be a center of rotational symmetry.

3. P and NP
P is the set of all yes/no problems which are decidable in polynomial time.
NP is the set of all yes/no problems in which a “proof” for a yes answer can be checked in polynomial time.
P is a subset of NP.
The question whether P=NP is one of the most important open questions in computer science.

4. NP-Completeness NP-complete problems are: A problem is NP-complete if:
it is in NP, and
the existence of a polynomial time algorithm to solve it implies the existence of a polynomial time algorithm for all problems in NP.
NP-complete problems are:
easy enough to check in polynomial time
the hardest such problems

5. Examples of NP-Completeness
Examples of NP-complete problems are:
3-Colorability: Can the vertices of a graph G be colored with 3 colors so that every pair of adjacent vertices has different colors?
Hamiltonicity: Does a graph G have a circuit that visits each vertex exactly once?
Bin Packing: Can we divide N numbers in K sets so that each set has sum less than S?
Satisfiability: Are there inputs to a Boolean circuit with AND/OR/NOT gates that make the outputs TRUE?

6. Spiral Galaxies Puzzles are NP-complete
The Main Result of this talk is:
The question of whether or not a given Spiral
Galaxies puzzle has a solution is NP-complete.
To prove this, we will build arbitrary Boolean circuits in the Spiral Galaxies universe.
"wires" carry truth values
"junctions" in wires simulate logical gates
Since Satisfiability is NP-complete, Spiral Galaxies puzzles are also NP-complete.

7. The Construction We need: wires
variables that can have either truth value
way to end a wire that forces it to be TRUE
NOT gate
AND gate
OR gate
way to split the signal in a wire
way to allow wires to cross

8. Wires and Signals
wires are rectangles of height 2 with a circle every 3 units.
a wire carries the value TRUE if the solution involves 3x2 rectangles and FALSE if the solution involves alternating 5x2 and 1x2 rectangles.
A TRUE signal
a FALSE signal

9. Variables variables are configurations with two local solutions.
A TRUE variable
a FALSE variable

10. Ending Wires
to force a TRUE or FALSE signal in a wire, we can end the wire at an appropriate point.
forcing a TRUE signal
forcing a FALSE signal

11. NOT Gate
the NOT gate is a wire that contains a pair of circles that are only 2 units away.

12. AND Gate

13. OR Gate

14. Letting Wires Cross

15. Splitting a Signal

16. Moving Wires
to shift a wire one unit left, we use three consecutive circles each a distance of 2.5 units from the previous one.
to shift a wire one unit up, we use three consecutive circles each raised .5 units from the previous one.

17. Filling in the Holes
to make the puzzle rectangular, we put a circle in every grid square that is not a part of the circuit.

18. Summary
For any given circuit, we can find a Spiral Galaxies puzzle that can be solved if and only if there is a set of inputs to the circuit that make the output TRUE.
This Satisfiability problem for circuits is known to be NP-complete.
The mapping we gave is polynomial.
Therefore whether or not a given Spiral Galaxies puzzle has a solution is also NP-complete.

19. References
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[3] E. D. Demaine and M. Hoffman, "Pushing blocks is NP-complete for non-crossing solution paths". Proc. 13th Canad. Conf. Comput. Geom. (2001),
[4] E. Friedman, "Corral Puzzles are NP-complete". preprint.
[5] E. Friedman, "Cubic is NP-complete". Proc Fl. Sectional MAA meeting, David Kerr ,Ed.
[6] E. Friedman, ”Pearl Puzzles are NP-complete". preprint.
[7] M.R. Garey and D.S. Johnson, Computers and Intractibility: A Guide to the Theory of NP-Completeness. W.H. Freeman, 1979.
[8] R. Kaye, "Minesweeper is NP-complete". Mathematical Intelligencer, 22 (2000) 9-15.
[9] Nikoli, 91 (2000).
[10] D. Ratner and M. Warmuth, "Finding a shortest solution for the n x n extension of the 15-puzzle is intractable". J. Symb. Comp. 10 (1990)
[11] S. Takahiro, "The Complexities of Puzzles, Cross Sum, and Their ASPs". preprint.
[12] Y. Takayuki, "On the NP-completeness of the Slither Link Puzzle". IPSJ SIGNotes Algorithms (2000).
[13] N. Ueda and T. Nagao, "NP-completeness Results for Nonogram via Parsimonious Reductions". preprint.