Expressing an NP-Complete Problem as the Solvability of a Polynomial Equation

We demonstrate a polynomial approach to express the decision version of the directed Hamiltonian Cycle Problem (HCP), which is NP-Complete, as the Solvability of a Polynomial Equation with a constant number of variables, within a bounded real space. We first introduce four new Theorems for a set of periodic Functions with irrational periods, based on which we then use a trigonometric substitution, to show how the HCP can be expressed as the Solvability of a single polynomial Equation with a constant number of variables. The feasible solution of each of these variables is bounded within two real numbers. We point out what future work is necessary to prove that P=NP.

💡 Research Summary

The paper under review attempts to bridge the gap between combinatorial NP‑complete problems and algebraic decision problems by showing how the directed Hamiltonian Cycle Problem (HCP) can be encoded as the solvability of a single polynomial equation with a constant number of variables, all confined to a bounded region of the real line. The authors begin by introducing four new theorems concerning families of periodic functions whose periods are irrational numbers. These theorems claim that for any two such functions, there exist arbitrarily small intervals in which both functions simultaneously attain zero, and that the set of such simultaneous zeros is dense. The purpose of these results is to provide a mathematical foundation for “compressing” many graph vertices into a few real variables using the phase of trigonometric functions with irrational frequencies.

The core construction proceeds as follows. Given a directed graph G = (V, E) with |V| = n, each vertex v_i is associated with a real angle θ_i. An edge (i → j) is represented by the term sin(θ_i – θ_j). The existence of a Hamiltonian cycle is then expressed by two families of constraints: (1) a “tour” constraint that forces each vertex to be visited exactly once, which the authors encode as a sum of squared sine terms that must equal zero, i.e., Σ_{k=1}^{n} sin²(θ_k – θ_{k+1}) = 0 (with θ_{n+1} ≡ θ_1); and (2) a “non‑edge” constraint that requires sin(θ_i – θ_j) ≠ 0 for every ordered pair (i, j) that is not an edge. By exploiting the irrational‑period theorems, the authors argue that the entire set of θ_i can be represented by a constant‑size vector (u_1,…,u_k), where k does not depend on n, because the dense zero set of the irrational‑period functions allows many angles to be synchronized without loss of information.

To obtain a purely algebraic formulation, the trigonometric terms are rewritten using Euler’s formula, turning each sin(θ_i – θ_j) into a rational combination of exponentials e^{iθ_i} and e^{-iθ_j}. After clearing denominators and eliminating complex parts, the authors arrive at a single multivariate polynomial Q(u_1,…,u_k) = 0 whose coefficients are built from the adjacency matrix of G. Importantly, they claim that each variable u_i is guaranteed to lie within a fixed interval