Approximate and Randomized Algorithms for Computing a Second Hamiltonian Cycle

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 deg...

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Vydáno v:Algorithmica Ročník 86; číslo 9; s. 2766 - 2785
Hlavní autoři: Deligkas, Argyrios, Mertzios, George B., Spirakis, Paul G., Zamaraev, Viktor
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.09.2024
Springer Nature B.V
Témata:
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
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Shrnutí: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.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-024-01238-z