置换Demarkko矩阵的识别
Recognising permuted Demidenko matrices
摘要：
我们解决了识别问题（RP）的类的Demerkko矩阵。我们的结果关闭了一个显着的差距，在识别特殊结构的矩阵。事实上，置换Dembruko矩阵的识别是一个长期的开放问题，相反，有效地解决RP的重要子类Dembruko矩阵，如卡尔曼森矩阵，Supnick矩阵，蒙格矩阵和反罗宾逊矩阵。置换Dembranko矩阵的识别是相关的硬组合优化问题的背景下，变得易于处理，如果输入是Dembranko矩阵。德姆林柯矩阵是德姆林柯在1976年提出的，当时他证明了如果对称距离矩阵满足某些组合条件（现在称为德姆林柯条件），则旅行商问题（TSP）是多项式可解的。在TSP的上下文中，识别问题在于决定是否存在城市的重新编号，使得相应重新编号的距离矩阵满足Demarkko条件，从而导致TSP的多项式可解的特殊情况。我们表明，这样的重新编号的n个城市可以找到在$O(n^4)$时间，如果它存在。
英文摘要：
We solve the recognition problem (RP) for the class of Demidenko matrices. Our result closes a remarkable gap in the recognition of specially structured matrices. Indeed, the recognition of permuted Demidenko matrices is a longstanding open problem, in contrast to the effciently solved RP for important subclasses of Demidenko matrices such as the Kalmanson matrices, the Supnick matrices, the Monge matrices and the Anti-Robinson matrices. The recognition of the permuted Demidenko matrices is relevant in the context of hard combinatorial optimization problems which become tractable if the input is a Demidenko matrix. Demidenko matrices were introduced by Demidenko in 1976, when he proved that the Travelling Salesman Problem (TSP) is polynomially solvable if the symmetric distance matrix fulfills certain combinatorial conditions, nowadays known as the Demidenko conditions. In the context of the TSP the recognition problem consists in deciding whether there is a renumbering of the cities such that the correspondingly renumbered distance matrix fulfills the Demidenko conditions, thus resulting in a polynomially solvable special case of the TSP. We show that such a renumbering of n cities can be found in $O(n^4)$ time, if it exists.
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