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Finding the symmetry group of an LP with equality constraints and its application to classifying orthogonal arrays
Discrete Optimization ( IF 1.1 ) Pub Date : 2019-01-28 , DOI: 10.1016/j.disopt.2019.01.001
Andrew J. Geyer , Dursun A. Bulutoglu , Kenneth J. Ryan

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 GLP, is the symmetry group of the LP. Margot (2010) described a method for computing a subgroup of the symmetry group GLP of an LP. This method computes GLP 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 GLP 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 (Margot, 2007). 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 (Bulutoglu and Ryan, 2018). These were the two bottleneck cases that could not have been solved until the B&B with isomorphism pruning algorithm was applied. Another key finding of this paper is that converting inequalities to equalities by introducing slack variables and exploiting the symmetry group of the resulting ILP’s LP relaxation within the B&B with isomorphism pruning algorithm can reduce the computation time by several orders of magnitude when enumerating a set of all non-isomorphic solutions of an ILP.



中文翻译:

求等式约束的LP对称群及其在正交阵列分类中的应用

对于给定的线性程序(LP),将可行点发送到可行点并保留其每个可行点的目标函数值的变量的排列是LP的对称性。LP的所有对称性的集合,表示为GLP,是LP的对称群。Margot(2010)描述了一种计算对称组子组的方法GLPLP。该方法计算GLP当LP仅具有非冗余不等式且其可行集不满足相等约束时。但是,当LP的可行集满足等式约束时,此方法仅找到GLP 并且会错过对称性。我们开发了一种方法,用于找到可行集满足相等约束的可行LP的对称群。我们应用这种方法来发现和利用同构修剪算法在分支定界(B&B)中定义整数线性程序(ILP)的正交数组的先前未利用的对称性(Margot,2007)。我们的方法减少了查找OA所有OD等效类的运行时间160824 和OA 176824 通过因素 1个2161个1个36与最快的已知方法相比(Bulutoglu和Ryan,2018)。这是在应用具有同构修剪的B&B之前无法解决的两个瓶颈情况。本文的另一个关键发现是,通过引入松弛变量并利用同构修剪算法在B&B中利用所得的ILP LP弛豫的对称组,将不等式转换为相等,在列举一组全部时,可以将计算时间减少几个数量级。 ILP的非同构解。

更新日期:2019-01-28
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