P-matrix recognition is co-NP-complete

This is a summary of the proof by G.E. Coxson that P-matrix recognition is co-NP-complete. The result follows by a reduction from the MAX CUT problem using results of S. Poljak and J. Rohn.

💡 Research Summary

The paper establishes that the decision problem “Is a given rational square matrix a P‑matrix?” is co‑NP‑complete. A P‑matrix is defined as a real square matrix all of whose principal minors are positive. The proof proceeds through a chain of four polynomial‑time reductions, each linking the target problem to a well‑known NP‑complete problem.

1. From SIMPLE MAX CUT to MATRIX R‑NORM.
   The authors start with the NP‑complete SIMPLE MAX CUT problem: given an undirected graph G=(V,E) and an integer K, decide whether there exists a cut of size at least K. They construct a matrix A = ℓ·I – A(G), where A(G) is the adjacency matrix of G and ℓ = 2|E|+1. Because ℓ dominates the off‑diagonal entries, A is strictly diagonally dominant and therefore nonsingular. For any vector y∈{−1,1}ⁿ they define S = {u : y_u = 1} and show that yᵀAy = nℓ + 4·|cut(S)| – 2|E|. Consequently, a cut of size at least K exists iff the r‑norm of A, defined as r(A)=max_{z,y∈{−1,1}ⁿ} zᵀAy, satisfies r(A) ≥ nℓ – 2|E| + 4K. This yields a polynomial‑time many‑one reduction from MAX CUT to the decision version of the r‑norm problem (Matrix R‑NORM), proving the latter NP‑hard. Moreover, because the matrix A is nonsingular, the problem remains NP‑complete even under the restriction to nonsingular inputs.

2. From MATRIX R‑NORM to RK1‑MATRIX‑INTERVAL‑SINGULARITY.
   The second reduction introduces the notion of a matrix interval