The Complexity of Checking Partial Total Positivity

TL;DR

This paper investigates the computational complexity of verifying partial total positivity in matrices, revealing co-NP-completeness in general cases but identifying polynomial-time solvability when the number of unspecified entries is logarithmic.

Contribution

It establishes the co-NP-completeness of checking partial total positivity and sign regularity, and shows polynomial-time solvability for matrices with few unspecified entries.

Findings

Checking partial total positivity is co-NP-complete.

Checking partial sign regularity is co-NP-complete.

Polynomial-time algorithm exists when unspecified entries are logarithmic in number.

Abstract

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.