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.

翻译：我们解决了德米登科矩阵类别的识别问题（RP）。这一成果填补了特殊结构矩阵识别领域长期存在的显著空白。事实上，与卡尔曼森矩阵、苏普尼克矩阵、蒙日矩阵和反罗宾逊矩阵等重要子类的高效识别形成鲜明对比，置换德米登科矩阵的识别是一个长期未决的开放问题。该识别问题在硬组合优化问题中具有重要意义——当输入为德米登科矩阵时，此类问题将变得可解。德米登科矩阵由Demidenko于1976年提出，其证明了若对称距离矩阵满足特定组合条件（现称德米登科条件），则旅行商问题（TSP）可在多项式时间内求解。在TSP背景下，识别问题在于判定是否存在一种城市重新编号方式，使得重排后的距离矩阵满足德米登科条件，从而将TSP转化为多项式可解的特例。我们证明：若存在此类重排，则可在$O(n^4)$时间内找到n个城市的这种重新编号。