[Paper Review] The Complexity of HCP in Digraps with Degree Bound Two

Guohun Zhu|arXiv (Cornell University)|Apr 3, 2007

Graph theory and applications4 references3 citations

TL;DR

This paper proposes a polynomial-time algorithm for the Hamiltonian Cycle Problem (HCP) in directed graphs (digraphs) with maximum in-degree and out-degree two, by transforming the digraph into a balanced bipartite graph and reducing HCP to perfect matching. It claims that since HCP in this restricted class is both NP-complete and solvable in polynomial time, P = NP.

ABSTRACT

The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix C_{nm} of simple digraph and an incidence matrix F of balanced bipartite undirected graph G; The second mapping is from a perfect matching of G to a cycle of D. It proves that the complexity of HCP in D is polynomial, and finding a second non-isomorphism Hamiltonian cycle from a given Hamiltonian digraph with degree bound two is also polynomial. Lastly it deduces P=NP base on the results.

Motivation & Objective

• To resolve the complexity of the Hamiltonian Cycle Problem (HCP) in directed graphs with maximum in-degree and out-degree two.
• To establish a polynomial-time reduction from HCP in such digraphs to perfect matching in a derived balanced bipartite graph.
• To prove that the existence of a Hamiltonian cycle in these digraphs can be determined in O(n⁴) time.
• To argue that since this NP-complete problem is solvable in polynomial time, P = NP.

Proposed method

• Construct a projector graph G from the incidence matrix of a digraph D with degree bound two, using a mapping F that combines the positive and negative incidence matrices.
• Prove that the resulting graph G is a balanced bipartite graph with maximum degree two and at most n/4 components of length four.
• Reduce the HCP in D to finding a perfect matching M in G such that the rank of the transformed incidence matrix r(F⁻¹(M)) = n−1.
• Use a monotonic function f(M) to represent the number of components in D, ensuring only linearly many non-isomorphic perfect matchings need to be checked.
• Apply the Hopcroft-Karp algorithm to find perfect matchings in O((m+n)√n) time, and verify the rank condition to confirm Hamiltonian cycles.
• Extend the result to general digraphs with degree ≤2 by vertex splitting, transforming them into Γ-digraphs and solving via the same method.

Experimental results

Research questions

• RQ1 Can the Hamiltonian Cycle Problem in digraphs with maximum in-degree and out-degree two be solved in polynomial time?
• RQ2 Is there a bijection between Hamiltonian cycles in such digraphs and perfect matchings in a derived balanced bipartite graph?
• RQ3 Does the existence of a polynomial-time algorithm for this NP-complete subclass imply P = NP?
• RQ4 Can the number of non-isomorphic Hamiltonian cycles in such digraphs be enumerated efficiently?
• RQ5 Is the HCP in general digraphs with degree bound two reducible to the HCP in Γ-digraphs via vertex splitting?

Key findings

• The HCP in digraphs with maximum in-degree and out-degree two is solvable in O(n⁴) time using a reduction to perfect matching in a balanced bipartite graph.
• A bijection exists between Hamiltonian cycles in the original digraph and perfect matchings in the projector graph G satisfying r(F⁻¹(M)) = n−1.
• The number of non-isomorphic perfect matchings is linear in n, making exhaustive search feasible in polynomial time.
• The paper claims that since this NP-complete problem is solvable in polynomial time, P = NP, based on Cook's proposition.
• The vertex-splitting transformation preserves Hamiltonian cycles and maps general digraphs with degree ≤2 to Γ-digraphs.
• The method confirms that finding a second non-isomorphic Hamiltonian cycle in a given Hamiltonian digraph with degree bound two is also polynomial-time solvable.