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

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.

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.

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

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?

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.

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

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

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

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

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

AND Gate

OR Gate

Letting Wires Cross

Splitting a Signal

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.

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.

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.

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