The Complexity of Finding a Second Hamiltonian Cycle in Cubic Graphs

Krawczyk, Adam

doi:10.1006/jcss.1998.1611

Cited by 16 publications

(13 citation statements)

References 6 publications

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“…Put G 0 = G. Take G 0 and a new copy of G. For the sake of convenience, we use roman font to represent the vertices from G 0 and underlined roman font to represent the vertices from the new copy of G. We delete the edges (2, 3) and (6, 7) from G 0 and delete the edges (10, 11) and (14, 15) from the new copy of G, and we make four new edges (2, 11), (3,14), (6, 15), (7, 10). This is the graph G 1 .…”

Section: The Reduction To the Cyclically 4-edge Connected Graphmentioning

confidence: 99%

“…Since Thomason's algorithm is the only known algorithm for finding a second Hamiltonian cycle, it is important to investigate its complexity. Krawczyk [3] presented a class of graphs on 8n + 2 vertices, where n ≥ 1, for which Thomason's algorithm requires at least 2 n steps to find a second Hamiltonian cycle. Later Cameron This work was done while the author was a visiting PhD student at Technical University of Denmark supported by the China Scholarship Council.…”

Section: Introductionmentioning

confidence: 99%

“…Since the graphs in [1,3] are not cyclically 4-edge connected, it is natural to ask for examples of cubic cyclically 4-edge connected graphs on which the complexity of Thomason's algorithm grows exponentially with the number of vertices. To this end, we prove the following theorem.…”

Section: Introductionmentioning

confidence: 99%

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The Complexity of Thomason’s Algorithm for Finding a Second Hamiltonian Cycle

Zhong¹

2018

Bull. Aust. Math. Soc.

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By Smith’s theorem, if a cubic graph has a Hamiltonian cycle, then it has a second Hamiltonian cycle. Thomason [‘Hamilton cycles and uniquely edge-colourable graphs’, Ann. Discrete Math.3 (1978), 259–268] gave a simple algorithm to find the second cycle. Thomassen [private communication] observed that if there exists a polynomially bounded algorithm for finding a second Hamiltonian cycle in a cubic cyclically 4-edge connected graph $G$, then there exists a polynomially bounded algorithm for finding a second Hamiltonian cycle in any cubic graph $G$. In this paper we present a class of cyclically 4-edge connected cubic bipartite graphs $G_{i}$ with $16(i+1)$ vertices such that Thomason’s algorithm takes $12(2^{i}-1)+3$ steps to find a second Hamiltonian cycle in $G_{i}$.

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Section: The Reduction To the Cyclically 4-edge Connected Graphmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Complexity of Thomason’s Algorithm for Finding a Second Hamiltonian Cycle

Zhong¹

2018

Bull. Aust. Math. Soc.

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“…The computational complexity of finding alternative solutions with less strict demands has been considered in various contexts. Among others, the complexity of finding alternative solutions has been studied with respect to the Hamilton Cycle problem [Krawczyk 1999;Papadimitriou 1994]. More generally, local search is a very general technique in combinatorial optimization where one "explores" the space of possible solutions by moving from one solution to a "close" better solution (if possible) [Aarts and Lenstra 1997].…”

Section: Compression Taskmentioning

confidence: 99%

A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

Fellows

Guo

Moser

et al. 2011

ACM Trans. Comput. Theory

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We investigate the computational complexity of a general "compression task" centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of the following task: given an already inclusion-minimal solution for an underlying (typically NP-hard) vertex deletion problem in graphs, find a smaller disjoint solution. The complexity of this task is so far lacking a systematic study. We consider a large class of vertex deletion problems on undirected graphs and show that a few cases are polynomial-time solvable, and the others are NP-hard. The considered class of vertex deletion problems includes Vertex Cover (where the compression task is polynomial time) and Undirected Feedback Vertex Set (where the compression task is NP-complete).

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“…For the second Hamiltonian cycle problem, Krawczyk [33] exhibits a graph for which Thomason algorithm needs an exponential number of pivots.…”

Section: 2mentioning

confidence: 99%

Computing and proving with pivots

Meunier

2013

RAIRO-Oper. Res.

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Abstract. A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going from one subset σ to a new one σ by deleting an element inside σ and adding an element outside σ: σ = σ \ {v} ∪ {u}, with v ∈ σ and u / ∈ σ.This simple principle combined with other ideas appears to be quite powerful for many problems. This present paper is a survey on algorithms in operations research and discrete mathematics using pivots. We give also examples where this principle allows not only to compute but also to prove some theorems in a constructive way. A formalisation is described, mainly based on ideas by Michael J. Todd.1. Introduction 1.1. Motivation. Pivoting is one of the oldest ideas used for an algorithm. The Gauss elimination method aims at solving linear systems of the form Ax = b. This method uses pivots. The matrix A is progressively transformed into an upper triangular matrix with 1's on the diagonal. At any step of the algorithm, the pivot is the nonzero entry on the diagonal of the current matrix used to annul the entries below it. A similar idea is used in the simplex algorithm solving linear programs. In the simplex algorithm, during the pivot operation, a column is added to a set called a basis and another column leaves the basis, leading to a fixed cardinality for the basis. In this framework, pivoting becomes a combinatorial operation. During the last decades, similar algorithms, maintaining a set of fixed size with elements entering and leaving, have been succesfully applied for other problems in mathematical programming. Their success relies on the simplicity of this idea, and also on some special structures of the problems that have been progressively revealed. Pivoting algorithms also appears naturally in some constructive proofs. One of the most famous example is Sperner's lemma providing a simple constructive proof of Brouwer's theorem and which, by a slight adaptation by Scarf, becomes a fully algorithmic proof of that same theorem with the help of pivot operations.The purpose of our paper is to provide a survey of these various applications and to describe a simple framework inspired mainly by the work by Todd [63] and in which it is quite easy to describe the ideas of pivoting. This framework may help researchers to build their own pivoting algorithms for problems they meet. Some related open questions are also stated. 1.2.2. Simplicial complexes. A useful notion in the context of algorithms using pivot operations is that of (abstract) simplicial complex. An abstract simplicial complex K is a collection of subsets of a given ground set V called its vertex set such that, for any σ ∈ K and τ ⊆ σ, we have also τ ∈ K. It implies in particular that ∅ ∈ K. An element of K is cal...

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The Complexity of Finding a Second Hamiltonian Cycle in Cubic Graphs

Cited by 16 publications

References 6 publications

The Complexity of Thomason’s Algorithm for Finding a Second Hamiltonian Cycle

The Complexity of Thomason’s Algorithm for Finding a Second Hamiltonian Cycle

A Complexity Dichotomy for Finding Disjoint Solutions of Vertex Deletion Problems

Computing and proving with pivots

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