破坏行列对称静态约束的复杂性和完备性
On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry
摘要：
我们考虑一种常见的对称类型，其中我们有一个矩阵的决策变量可互换的行和列。处理这种行和列对称性的一种简单而有效的方法是发布对称性破坏约束，如DOUBLELEX和SNAKELEX。我们提供了一些积极的和消极的结果张贴这样的对称性破缺的限制。在积极的一面，我们证明，我们可以计算在多项式时间的唯一代表的等价类的矩阵模型的行列对称，如果行（或列）的数量是有界的，并在一些其他特殊情况下。在消极的一面，我们表明，虽然DOUBLELEX和SNAKELEX在实践中往往是有效的，他们可以留下大量的对称解决方案在最坏的情况下。此外，我们证明了完全传播DOUBLELEX是NP-困难的。最后，我们考虑如何打破行，列和值的对称性，纠正了文献中的结果，结合不同的对称性破缺约束的安全性。最后，我们的第一个实验研究有多少对称性留下的DOUBLELEX和SNAKELEX的一些基准问题。
英文摘要：
We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DOUBLELEX completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
