Erich FriedmanStetson UniversityOctober 2, 2002

Spiral Galaxies Puzzlesare NP-complete

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-CompletenessA 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

forcing a TRUE signal forcing a FALSE signal

• to force a TRUE or FALSE signal in a wire, we can end the wire at an appropriate point.

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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