We prove that checking if a partial matrix is partial totally positive is co-NP-complete. This contrasts with checking a conventional matrix for total positivity, for which we provide a cubic time algorithm. Checking partial sign regularity with any signature, including partial total nonnegativity, is also co-NP-complete. Finally, we prove that checking partial total positivity in a partial matrix with logarithmically many unspecified entries may be done in polynomial time.
翻译:我们证明检查一个部分矩阵是否完全正数是完全正数是共同NP完成的。这与检查一个常规矩阵是完全正数的对比,我们为此提供了一种立方时间算法。检查任何签名的局部符号规律性,包括部分非全数,也是共同NP完成的。最后,我们证明在一个部分矩阵中检查部分完全假设性,加上对数性的许多未具体说明的条目,可以在多符号时间内完成。