The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore–Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account.

中文翻译：

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二次循环覆盖问题：特殊情况和有效边界

二次循环覆盖问题是找到一组访问所有节点的节点不相交循环的问题，从而使连续弧之间的交互成本的总和最小化。在本文中，我们研究了二次循环覆盖问题和相关下界的线性化问题。特别是，我们得出了使二次成本矩阵可线性化的各种充分条件，并使用这些条件来计算界限。我们还展示了如何在迭代包围过程中为线性化使用足够的条件。在每个步骤中，我们的算法都会针对给定的线性化条件，计算二次成本矩阵及其最佳线性化矩阵的最佳等效表示形式。进一步，我们证明了经典的Gilmore-Lawler类型的边界属于基于线性化的边界的族，因此应用了上述迭代重构技术。我们还证明了这种迭代方法得到的线性化矢量满足常数值的性质。考虑到质量和效率，此处介绍的界限中最好的界限优于现有的界限。