In my research, I need to find the set of all minimal dependent sets of a given set of vectors. One method is to check every subset of the given set. But this method is very slow when the set of vectors is large. For example, let $S$ be the set of all positive roots of type $B_5$ root system. Then $S$ consists of the following vectors \begin{align} &a_5, \quad a_4+2 a_5, \quad a_4+a_5, \quad a_3+a_4+2 a_5, \quad a_3+2 a_4+2 a_5, \\ &a_3+a_4+a_5, \quad a_2+a_3+a_4+2 a_5, \quad a_2+a_3+2 a_4+2 a_5, \quad a_2+2 a_3+2 a_4+2 a_5, \\ &a_2+a_3+a_4+a_5, \quad a_1+a_2+a_3+a_4+2 a_5, \quad a_1+a_2+a_3+2 a_4+2 a_5, \\ &a_1+a_2+2 a_3+2 a_4+2 a_5, \quad a_1+2 a_2+2 a_3+2 a_4+2 a_5, \quad a_1+a_2+a_3+a_4+a_5, \\ &a_4, \quad a_3+a_4, \quad a_2+a_3+a_4, \quad a_1+a_2+a_3+a_4, \quad a_3, \\ &a_2+a_3, \quad a_1+a_2+a_3, \quad a_2, \quad a_1+a_2, \quad a_1, \end{align} where $a_i$'s are simple roots.

I let the computer run for one day but didn't get the result. Is there some effective method to compute all minimal dependent sets of a given set of vectors? Could this be done in Sage or Maple or other software? Thank you very much.

allminimal circuits in polynomial time rather than searching all possible subsets. $\endgroup$ – François G. Dorais♦ Sep 30 at 14:49