Sign-consistency and solvability of constrained linear systems

Abstract

Abstract. Sign-solvable linear systems were introduced in modelling economic and physical systems where only qualitative information is known. Often economic and physical constraints require the entries of a solution to be nonnegative. Yet, to date the assumption of nonnegativity has been omitted in the study of sign-solvable linear systems. In this paper, the notions of signconsistency and sign-solvability of a constrained linear system Ax = b; x 0 0, are introduced. These notions give rise to new classes of sign patterns. The structure and the complexity of the recognition problem for each of these classes are studied. A qualitative analog of Farkas' Lemma is proven, and it is used to establish necessary and su cient conditionsfor the constrainedlinear system Ax = b; x 0 0 to be sign-consistent. Also, necessary and su cient conditions for the constrained linear system Ax = b; x 0 0 to be sign-solvable are determined, and these are used to establish a polynomialtime recognition algorithm. It is worth noting that the recognition problem for unconstrained sign-solvable linear systems is known to be NP-complete.

Strict Sign-Central Matrices

“…In [9], the following notion was introduced. A (0, ±1)-matrix A is called an L + -matrix provided that each B ∈ Q(A) satisfies {y : y T B ≥ O} = {O}.…”

Sign patterns that require a positive or nonnegative left inverse

“…x > 0) is a constrained system, and the constrained system is sign-consistent provided, for each matrix A ∈ Q(A) and each vector b ∈ Q( b), there exists an m by 1 nonzero and nonnegative (resp. positive) vector x such that x T A = b T ; see [5]. Note that, for an n by m matrix X with rows x T i and an m by n matrix A, XA = I n implies that x…”

“…We denote by e i the column sign pattern with a + entry in row i and 0's elsewhere. Using this observation and results on sign-consistent, constrained systems in [5], in Section 3, we give necessary and sufficient conditions for an m by n sign pattern to require an NLI or a PLI.…”

“…General Case. In this section, by using results on sign-consistent, constrained systems in [5], we give necessary and sufficient conditions for a sign pattern to require an NLI or a PLI.…”

“…Hence, transposes of S-matrices provide a family of sign patterns which require a PLI. Note that S-matrices can be recognized in polynomial-time (see [3]), while the problem of recognizing if a sign pattern is not an L + -matrix is NP-complete (see [5]). Recall that there are no square sign patterns of order n with n ≥ 2 that require a PLI (see Theorem 2.1), but there are square sign patterns of order n with n ≥ 2 that allow a PLI (see [2, Theorem 9.2.1]).…”