Add code to generate supernecklaces and -bracelets

[pozorvlak]

A necklace is an equivalence class of strings under rotation; a bracelet is an equivalence class of strings under rotation and reflection. Restricting our attention further to necklaces/bracelets without
repetitions, we can search for "supernecklaces" and "superbracelets", which are strings that contain some representative of each non-repeating necklace/bracelet as a substring (equivalently, strings
which, for every permutation p, contain a substring that may be rotated and/or reflected to obtain p).

This is not a classic TSP, but it can be framed as an Equality-Generalised Travelling Salesman Problem, which can then be converted into an ATSP and solved by the GLKH solver, http://webhotel4.ruc.dk/~keld/research/GLKH/.

This work was inspired by Matthew Clarke's questions on the forum: https://groups.google.com/forum/?#!msg/superpermutators/H1_jrq_EO1c/unrp4Gl_GQAJ

I don't think this is quite ready to be merged: I'd like to make it a bit easier to actually run the solver first (I had to hack up the supplied runGLKH script).

[pozorvlak]

More seriously, it's not finding the shortest supernecklace for n = 3: I'm getting 123213, but unless I'm missing something then 12321 should do. Here's my input file:

NAME: supernecklace 3
TYPE: GTSP
DIMENSION : 6
EDGE_WEIGHT_TYPE : EXPLICIT
EDGE_WEIGHT_FORMAT : FULL_MATRIX
NODE_COORD_TYPE : NO_COORDS
DISPLAY_DATA_TYPE : NO_DISPLAY
EDGE_WEIGHT_SECTION :
0 3 3 1 2 2
0 0 2 2 3 1
0 1 0 3 2 2
0 2 3 0 1 3
0 3 2 2 0 3
0 2 1 3 3 0
GTSP_SETS: 2
GTSP_SET_SECTION
1 1 4 5 -1
2 2 3 6 -1
EOF

and here's the output file:

NAME : supernecklace.0.tour
COMMENT : Length = 0
COMMENT : Found by GLKH [Keld Helsgaun] Sun Feb 17 22:15:58 2019
TYPE : TOUR
DIMENSION : 2
TOUR_SECTION
1
3
-1
EOF

and here's the .par file I used (generated by the runGLKH script that came with GLKH):

PROBLEM_FILE = gtsp/necklace_3.gtsp
ASCENT_CANDIDATES = 500
INITIAL_PERIOD = 1000
MAX_CANDIDATES = 30
MAX_TRIALS = 1000
OUTPUT_TOUR_FILE = gtsp/necklace_3.gtsp.$.tour
PI_FILE = PI_FILES/gtsp/necklace_3.gtsp.pi
POPULATION_SIZE = 1
PRECISION = 10
RUNS = 1
SEED = 1
TRACE_LEVEL = 1

Am I doing something obviously wrong? I couldn't immediately find a spec for the .gtsp format, so I reverse-engineered it from the supplied examples. Am I right in thinking that LKH isn't guaranteed to find the shortest possible tour? Even so, I'd expect it to do so on such a small example.

[pozorvlak]

I also wrote some much more naive code for this problem in the constraint-solving language MiniZinc: it's vastly slower than this, but does at least get the right answer for the n = 3 supernecklace. Would it be welcome in this repository?

[robinhouston]

I won’t lie: the most exciting thing about this for me is that I have learnt about GitHub’s new “draft pull request” feature! :-)

[robinhouston]

@pozorvlak The MiniZinc code would be very welcome, yes! We are language-agnostic here. :-)

[pozorvlak]

I fixed the problem with finding overlong supernecklaces, by the way: GLKH was interpreting my input files as containing undirected graphs. The two-character fix is in d9de7cc.