We study the time complexity of computing the $(\min,+)$ matrix product of two $n\times n$ integer matrices in terms of $n$ and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most $m_1$ monotone subsequences and each column of the second matrix can be decomposed into at most $m_2$ monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the $(\min,+)$ product of the matrices can be computed in $O(m_1m_2n^{2.569})$ time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or {\em vice versa} then this case is as hard as the general one. Similarly, we also study the time complexity of computing the $(\min,+)$ convolution of two $n$-dimensional integer vectors in terms of $n$ and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most $m_1$ monotone subsequences and the second vector can be decomposed into at most $m_2$ subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or {\em vice versa} then their $(\min,+)$ convolution can be computed in $\tilde{O}(m_1m_2n^{1.5})$ time. On the other, the case when both vectors are non-decreasing or both of them are non-increasing is as hard as the general case.

翻译：我们研究两个n×n整数矩阵的(min,+)矩阵乘积的时间复杂度，其复杂度取决于n以及第一个矩阵行向量和第二个矩阵列向量可分解的单调子序列数量。具体而言，我们证明：若第一个矩阵的每一行可分解为至多m₁个单调子序列，且第二个矩阵的每一列可分解为至多m₂个单调子序列，且所有子序列均为非递减或均为非递增，则矩阵的(min,+)乘积可在O(m₁m₂n^{2.569})时间内计算。另一方面，我们观察到若第一个矩阵的所有行均为非递减且第二个矩阵的所有列均为非递增（或反之），则该情形与一般情形同样困难。类似地，我们还研究两个n维整数向量的(min,+)卷积的时间复杂度，其复杂度取决于n以及两个向量可分解的单调子序列数量。我们证明：若第一个向量可分解为至多m₁个单调子序列，第二个向量可分解为至多m₂个子序列，且第一个向量的所有子序列均为非递减、第二个向量的所有子序列均为非递增（或反之），则它们的(min,+)卷积可在Õ(m₁m₂n^{1.5})时间内计算。另一方面，当两个向量均为非递减或均为非递增时，该情形与一般情形同样困难。