Convex and linear models of NP-problems

Reducing the NP-problems to the convex/linear analysis on the Birkhoff polytope.

💡 Research Summary

The paper “Convex and linear models of NP‑problems” proposes a novel framework that translates decision‑type NP‑complete problems into feasibility problems on the Birkhoff polytope, the convex hull of all (n\times n) permutation matrices. The authors begin by observing that any solution to a combinatorial NP problem can be encoded as a permutation matrix (P). Since the set of all doubly‑stochastic matrices (\mathcal{B}_n={X\ge0\mid X\mathbf{1}=\mathbf{1}, X^{\top}\mathbf{1}=\mathbf{1}}) is exactly the convex hull of these permutation matrices, the existence of a solution is equivalent to the existence of a point in (\mathcal{B}_n) satisfying a collection of linear inequalities derived from the original instance.

The core contribution is a systematic, polynomial‑time reduction that maps an arbitrary NP decision problem (\Pi) with input (I) to a linear system \