Finding the symmetry group of an LP with equality constraints and its application to classifying orthogonal arrays

Abstract: For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by $G^{\rm LP}$, is the symmetry group of the LP. Margot [F. Margot, 50 Years of Integer Programming 1958-2008 (2010), 647-686] described a method for computing a subgroup of the symmetry group $G^{\rm LP}$ of an LP. This method computes $G^{\rm LP}$ when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints. However, when the feasible set of the LP satisfies equality constraints this method finds only a subgroup of $G^{\rm LP}$ and can miss symmetries. We develop a method for finding the symmetry group of a feasible LP whose feasible set satisfies equality constraints. We apply this method to find and exploit the previously unexploited symmetries of an orthogonal array defining integer linear program (ILP) within the branch-and-bound (B&B) with isomorphism pruning algorithm [F. Margot, Symmetric ILP: Coloring and small integers, Discrete Optimization 4 (1) (2007), 40-62]. Our method reduced the running time for finding all OD-equivalence classes of OA$(160,8,2,4)$ and OA$(176,8,2,4)$ by factors of $1/(2.16)$ and $1/(1.36)$ compared to the fastest known method [D. A. Bulutoglu and K. J. Ryan, Integer programming for classifying orthogonal arrays, Australasian Journal of Combinatorics 70 (3) (2018), 362-385]. These were the two bottleneck cases that could not have been solved until the B&B with isomorphism pruning algorithm was applied.