reductions

This is a Java program that implements reductions demonstrating that certain problems are hard. Specifically, it demonstrates that Mario, Zelda, Pokémon, Push-1, and PushPush are NP-hard using reductions from 3-SAT. This program also implements algorithms that are actually useful, including planarity testing, planar embedding, and graph drawing algorithms. It is tested in Java 7.0, but I think it probably works in Java 5.0 and higher.

Legitimately useful features

This project includes implementations of the following algorithms, each of which takes linear time assuming constant amortized HashMap / HashSet performance:

Testing whether a graph is planar. (A graph is planar if it has a planar drawing, which assigns each vertex a 2D point and draws the edges as non-crossing curves in the plane.)
Computing a planar embedding for a planar graph. (A planar embedding is a clockwise ordering of the edges around each vertex and an external face that are feasible in a planar drawing.)
Computing a weak visibility representation for a planar graph. (A weak visibility representation is a drawing of a planar graph where vertices are horizontal line segments and edges are vertical line segments, none of which cross.)
Computing an SPQR tree. (See src/com/github/btrekkie/graph/spqr/SpqrNode.java for a description of SPQR trees.)
Computing an EC-planar embedding, and testing for EC-planarity. An EC-planar embedding is a planar embedding with certain constraints on the clockwise ordering of the edges around the vertices.
Computing a block-cut tree. (See src/com/github/btrekkie/graph/bc/BlockNode.java for a description of block-cut trees.)
Computing a dual graph. (The dual H of a graph G is a graph with one vertex for each face in G, including the external face, and an edge between each pair of vertices corresponding to two faces in G separated by an edge.)
Augmenting a planar graph to be biconnected while preserving a planar embedding. (A graph is biconnected if it remains connected if we remove any vertex. Augmentation is the addition of edges.)

It also includes the following features:

Computing a set of edge crossings to add to a graph to make it planar or EC-planar. I haven't done a good analysis of the number of crossings the algorithm produces, but in the interest of full disclosure, I will say I get the sense that it's rather large.
A Swing JComponent and JFrame for displaying images rendered programatically. This is necessary for the reductions because some of the stages they produce are so large that storing the results as an image file is impractical. Instead, the program renders the stages at the desired position and level of zoom on the fly.

Fun features

This program demonstrates that the original Super Mario Bros., the original Legend of Zelda, all versions of Pokémon, and the Push-1 and PushPush games are NP-hard using reductions from 3-SAT. In each case, the problem is whether it is possible to get from here to there. For an explanation of the term "NP-hard", see the Wikipedia article on P vs. NP. Basically, any program that determines whether it is possible to beat a level in one of the aforementioned games must take a long time to run, in a certain formal sense. (This is assuming the unproven but widely believed claim that P ≠ NP.) For a definition of 3-SAT, see src/com/github/btrekkie/reductions/bool/ThreeSat.java.

Demo

For a demonstration of the Mario reduction, run the following UNIX commands from the root project directory:

mkdir bin
javac -sourcepath src `find . -name "*.java" | grep -v Test` -d bin
java -cp bin com.github.btrekkie.reductions.mario.MarioProblem

A window displaying a Mario level should appear. The level takes a while to render. For faster results, click the "+" button 15 to 20 times. To see demonstrations of the Zelda, Pokémon, or Push-1 / PushPush reductions, change mario.MarioProblem in the last command to zelda.ZeldaProblem, pokemon.PokemonProblem, or push1.Push1Problem respectively.

Pokémon problems feature strong trainers, who always defeat the player, and weak trainers, who never defeat the player. The sight lines of the strong trainers are show in blue, and the sight lines of the weak trainers are shown in red.

The gadgets and reduction structure are taken from Aloupis, Demaine, Guo, Viglietta (2012): Classic Nintendo Games are (Computationally) Hard and Demaine, Demaine, and O'Rourke (2000): PushPush and Push-1 are NP-hard in 2D. See those papers for a more detailed description of what is going on in the stages this program produces.

Documentation

See https://btrekkie.github.io/reductions/index.html for API documentation.

Credits

The Super Mario Bros. sprite sheet was composed by Beam Luinsir Yosho.
The Legend of Zelda sprites are from The Spriters Resource, ripped by Alien Link and by an unknown contributor.
The algorithms used for the reductions and for various graph drawing problems are taken or adapted from different papers and notes, credited in the comments for the respective classes and methods.

btrekkie / reductions