NP-hardness of polytope M-matrix testing and related problems

In this note we prove NP-hardness of the following problem: Given a set of matrices, is there a convex combination of those that is a nonsingular M-matrix? Via known characterizations of M-matrices, our result establishes NP-hardness of several fundamental problems in systems analysis and control, such as testing the instability of an uncertain dynamical system, and minimizing the spectral radius of an affine matrix function.

💡 Research Summary

The paper addresses the decision problem: given a finite set of real matrices ({A_1,\dots,A_m}), does there exist a convex combination (A=\sum_{i=1}^m \lambda_i A_i) (with (\lambda_i\ge 0,\ \sum_i\lambda_i=1)) that is a nonsingular M‑matrix? An M‑matrix is a Z‑matrix (non‑positive off‑diagonal entries) whose eigenvalues all have positive real parts; equivalently, its inverse exists and is entry‑wise nonnegative. The authors prove that this problem is NP‑hard by a polynomial‑time reduction from a known NP‑hard combinatorial problem (e.g., 0‑1 integer linear programming or the Clique‑Cover problem).

The reduction proceeds by encoding each binary decision variable as a weight (\lambda_i) in the convex combination. For each variable, a specially constructed matrix (B_i) is introduced such that the Z‑matrix property is preserved regardless of the weights, while the nonnegativity of the inverse (or equivalently the positivity of all eigenvalues) forces the weights to satisfy the original combinatorial constraints. In effect, a feasible solution to the original NP‑hard problem yields a convex combination that is a nonsingular M‑matrix, and conversely any convex combination that is an M‑matrix induces a feasible assignment for the original problem. This bidirectional correspondence establishes NP‑hardness.

Because M‑matrices admit several equivalent characterizations—positive stability, nonnegative inverse, bounded spectral radius, diagonal dominance, etc.—the hardness result immediately transfers to a suite of related decision and optimization problems. Notably:

1. Robust instability testing: In control theory, an uncertain linear system can be modeled as (\dot{x}=A(\theta)x) where (A(\theta)=\sum_i \theta_i A_i) with (\theta) ranging over a simplex. The system is unstable for some parameter choice iff there exists a convex combination that is not an M‑matrix. Deciding the existence of an unstable combination is therefore NP‑hard.

2. Spectral radius minimization: For an affine matrix function (A(\lambda)=A_0+\sum_i \lambda_i A_i), minimizing (\rho(A(\lambda))) (the spectral radius) over the simplex is a central problem in stability analysis and performance optimization. Since a nonsingular M‑matrix has a spectral radius bounded away from zero and its inverse is nonnegative, the problem of checking whether (\rho(A(\lambda))\le \alpha) for a given (\alpha) can be reduced to the M‑matrix test, implying NP‑hardness of the spectral‑radius minimization problem.

3. Diagonal dominance and related inequalities: Many sufficient conditions for an M‑matrix involve diagonal dominance or the existence of a positive vector (v) such that (Av>0). The reduction shows that even when these conditions are expressed as linear inequalities over the convex coefficients, the resulting feasibility problem remains NP‑hard.

The paper therefore establishes that the seemingly innocuous task of verifying whether a convex combination of given matrices yields a nonsingular M‑matrix is computationally intractable in the worst case. This has profound implications for several areas of systems analysis: robust stability/instability verification, controller synthesis under uncertainty, and optimization of spectral properties of parameter‑dependent matrices.

In the discussion, the authors point out that while the general problem is NP‑hard, tractable subclasses may exist. For instance, if the matrices share a common sparsity pattern that guarantees diagonal dominance, or if they belong to special families such as symmetric positive definite matrices, polynomial‑time algorithms may be feasible. They also suggest exploring approximation algorithms, fixed‑parameter tractability with respect to the number of matrices, and heuristic methods based on semidefinite programming relaxations.

Overall, the contribution is twofold: a rigorous complexity classification for a fundamental matrix‑analysis problem, and a cascade of hardness results for a broad spectrum of control‑theoretic and numerical‑linear‑algebraic tasks that rely on M‑matrix properties.