[GAP Forum] Question regarding testing lex min representatives of orbits

[jim ostrowski]

Hello, I am a graduate student in Industrial Engineering and I trying to
learn how to use GAP. I am still trying to improve my computational algebra,
so please excuse any misuses of notation.

I am trying to solve Integer programs of the form min e^t x    s.t.  Ax \geq
e, x in {0, 1}.  The matrix A has a lot of symmetry (permutations of the
columns s.t. there exists a permutation of the rows which send A to A). I am
trying to solve this problem by branch and bound. At each node in the
enumeration tree I have a set of variables, F,  fixed to 1. I need to check
if there exists a set F' in the orbit of F (w.r.t. a given symmetry group)
which is lexicographically smaller than F. For example, I have a group
consisting of the permutation (1,2). I need to test if the set (word) {2,3}
is a lex min representative of its orbit (which it is not, since {1,3} <
{2,3}).

Can I do this directly in GAP? I have tried to read through the
documentation, and I haven't seen anything that makes me think I can do it.
If not, is there any other software available which would do this? Would it
be difficult to write code in GAP which would do this?

Assuming I am able to do the above, I am next interested in testing for lex
min using a rank vector. So, using the above example, if the rank of the
element "2" is less than the rank of the element "1", then {2,3} < {1,3}.

Thank you for reading my question. I feel bad asking what I am sure is a
basic question, but I thought it wouldn't hurt to send this out before I
start having to code something from scratch.
~Jim Ostrowski