jcpaik / sofa-designer

A software that assists a prospected proof of Gerver's conjecture, using a custom geometric branch-and-bound algorithm, and the exact rational QP solver powered by CGAL

sofa-designer

The program is the computer-assited part of a prospected proof on the moving sofa problem in progress. It is developed by Jineon Baek.

An essential idea of Gerver is to see the movement in perspective of the moving sofa, so that the sofa is fixed and the hallway rotates and translates around it. Then the sofa is essentially the intersection of the hallways being rotated counterclockwise from 0 to 90 degrees and translated. The problem can be approximated by intersecting only a finite number of hallways in finite angles, and Gerver proved that the optimum to this approximation converges to the optimal sofa.

SofaDesigner does the followings.

1. It computes every possibility of the intersection of hallways rotated by a given finite list of angles.
2. It proves lower and upper bound on any linear functional on the locations of hallways by examining each possibility.

Usage

The software is still in development and the details are subject to change.

The angles are represented as a single JSON list.

[
    {"cos": "12/13", "sin": "5/13", "branch_order": 3},
    {"cos": "4/5", "sin": "3/5", "branch_order": 1},
    {"cos": "3/5"  , "sin": "4/5"  , "branch_order": 0},
    {"cos": "5/13"  , "sin": "12/13"  , "branch_order": 2}
]

The angles should be strictly in between 0 and 90 degrees and have rational cos and sin values. The field branch_order determines the order in which the hallway is included in the main branch-and-bound algorithm.

Once the program is built, do

./sbranch --angles angles.json --out angles.crl

to store every possibilities of the intersection of hallways rotated by angles in angles.json in a file angles.crl. The --show-max-area flag also computes the maximum possible area of such intersection, giving an upper bound of the area of any sofa rotating by 90 degrees.

Then, use the angles.crl file to prove lower/upper bound of any linear functional as the following.

./sprove angles.crl "dot(A(0)-A(5),u(0))" --lb 0 --ub 1

See runs/ directory for more examples.

Structure

lib contains main, common libraries developed for the project. Currently the only library is sofa.

bin contains source code for the final binaries sbranch and sprove.

test contains test codes based on catch2 library. A single executable run_tests runs all tests (except hidden tests).

extern contains external libraries the project depends on.

How to build

Install cmake and libraries gmp and cgal. This is

brew install cmake gmp cgal catch2

on MacOS. Check respective repository for other OS. I found successful compilation under the following versions.

• CGAL 5.5.2
• GMP 6.2.1_1
• CMake 3.26.3
• Catch2 3.4.0

To build, make and cd to the build directory and run cmake ..; make.

mkdir build
cd build
cmake ..
make

The binaries will appear in build/bin.

Acknowledgements

@jwvg0425 has helped in shaping the program architecture and making the code to follow better design patterns.