The Complexity of Determining Existence a Hamiltonian Cycle is $O(n^3)$

The Hamiltonian cycle problem in digraph is mapped into a matching cover bipartite graph. Based on this mapping, it is proved that determining existence a Hamiltonian cycle in graph is $O(n^3)$.

💡 Research Summary

The paper attempts to settle the long‑standing NP‑complete Hamiltonian‑cycle decision problem by translating a directed graph G(V,E) into a bipartite graph B and then solving a matching problem in B. The translation is straightforward: each vertex v∈V is duplicated into v_in and v_out; every directed edge (u→v) in G becomes an undirected edge (u_out , v_in) in B. The authors then introduce the notion of a “matching cover,” which they define as a set of edges that matches every vertex of B exactly once. They claim that the existence of such a matching cover is equivalent to the existence of a Hamiltonian cycle in the original digraph.

To support this claim, the paper outlines two algorithmic phases. In the first phase, a perfect matching (or matching cover) in B is found. The authors assert that a modified Hungarian algorithm can accomplish this in O(n²) time, where n = |V|. In the second phase, the matched edges are mapped back to directed edges of G, and the authors argue that these edges can be reordered into a single Hamiltonian cycle in O(n) time. Adding the two phases yields an overall time bound of O(n³), which the authors present as a breakthrough polynomial‑time solution for a problem traditionally believed to require exponential time unless P = NP.

A careful technical analysis reveals several fatal flaws. First, the equivalence between a matching cover in B and a Hamiltonian cycle in G is not rigorously proved. A perfect matching in B merely guarantees a collection of vertex‑disjoint paths or cycles when projected back onto G; there is no guarantee that these components can be merged into one simple cycle without additional constraints. The paper does not provide the necessary connectivity or ordering conditions that would enforce a single Hamiltonian tour.

Second, the claimed O(n²) complexity for finding a perfect matching is unsupported. The classic Hungarian algorithm runs in Θ(n³) for dense bipartite graphs, and faster algorithms (e.g., Hopcroft–Karp) achieve O(√n·m) where m is the number of edges. The authors do not specify the structure of B that would allow a universally O(n²) solution; indeed, for arbitrary digraphs B can be highly sparse or irregular, making the worst‑case matching time at least as large as the known bounds. Without a concrete algorithmic description and a proof that it works for all inputs, the O(n²) claim remains speculative.

Third, the overall conclusion that Hamiltonian‑cycle detection lies in O(n³) contradicts established complexity theory. The Hamiltonian‑cycle decision problem is NP‑complete, and a polynomial‑time algorithm for it would imply P = NP, a result that has eluded proof for decades. The paper does not address this implication, nor does it restrict its claims to a special subclass of digraphs (e.g., tournaments, planar graphs) where Hamiltonicity can be decided more efficiently. As presented, the algorithm purports to solve the general problem, which is inconsistent with the current understanding of computational complexity.

In summary, while the idea of reducing Hamiltonicity to a bipartite matching problem is conceptually appealing, the paper falls short on three critical fronts: (1) it lacks a rigorous proof of the bijective relationship between matching covers and Hamiltonian cycles; (2) it provides an unsubstantiated complexity analysis for the matching step; and (3) it overlooks the profound theoretical consequences of claiming a cubic‑time algorithm for an NP‑complete problem. Consequently, the central claim—that Hamiltonian‑cycle existence can be decided in O(n³) time—cannot be accepted without substantial revisions, detailed algorithmic specifications, and a clear delineation of the problem’s scope. Future work would need to either prove the missing equivalence, present a verifiable O(n³) matching algorithm for all bipartite graphs derived from digraphs, or explicitly limit the result to a tractable subclass of graphs.