Approximate and Randomized Algorithms for Computing a Second Hamiltonian Cycle

• Published in: Algorithmica Vol. 86; no. 9; pp. 2766 - 2785
• Main Authors: Deligkas, Argyrios, Mertzios, George B., Spirakis, Paul G., Zamaraev, Viktor
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2024; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computation; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Mathematics of Computing; Statistical analysis; Theory of Computation; Hamiltonian cycle; Approximation algorithm; Randomized algorithm; Graph with minimum degree 3
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-024-01238-z

In this paper we consider the following problem: Given a Hamiltonian graph G , and a Hamiltonian cycle C of G , can we compute a second Hamiltonian cycle C ′ ≠ C of G , and if yes, how quickly? If the input graph G satisfies certain conditions (e.g. if every vertex of G is odd, or if the minimum degree is large enough), it is known that such a second Hamiltonian cycle always exists. Despite substantial efforts, no subexponential-time algorithm is known for this problem. In this paper we relax the problem of computing a second Hamiltonian cycle in two ways. First, we consider approximating the length of a second longest cycle on n -vertex graphs with minimum degree δ and maximum degree Δ . We provide a linear-time algorithm for computing a cycle C ′ ≠ C of length at least n - 4 α ( n + 2 α ) + 8 , where α = Δ - 2 δ - 2 . This results provides a constructive proof of a recent result by Girão, Kittipassorn, and Narayanan in the regime of Δ δ = o ( n ) . Our second relaxation of the problem is probabilistic. We propose a randomized algorithm which computes a second Hamiltonian cycle with high probability , given that the input graph G has a large enough minimum degree. More specifically, we prove that for every 0 < p ≤ 0.02 , if the minimum degree of G is at least 8 p log 8 n + 4 , then a second Hamiltonian cycle can be computed with probability at least 1 - 1 n 50 p 4 + 1 in p o l y ( n ) · 2 4 p n time. This result implies that, when the minimum degree δ is sufficiently large, we can compute with high probability a second Hamiltonian cycle faster than any known deterministic algorithm. In particular, when δ = ω ( log n ) , our probabilistic algorithm works in 2 o ( n ) time.