A short note on graphs with long Thomason chains

Briański, Marcin; Szady, Adam

doi:10.1016/j.disc.2021.112624

Cited by 2 publications

(3 citation statements)

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“…A specific algorithm that has been frequently studied in this light is Thomasen's lollipop algorithm for finding a second Hamiltonian cycle in an (explicit) cubic graph by following a path in a much larger implicit graph defined from the given graph [50]. Although some inputs cause this algorithm to take exponential time [6,7,58] the complexity of finding a second Hamiltonian cycle in a different way is unknown, and was one of the motivating problems for the definition of PPA [44]. We formulate the same question in a different way, asking how hard it is to find the same cycle that Thomasen's algorithm finds, but again the complexity of this problem remains unknown.…”

Section: Related Workmentioning

confidence: 99%

The Complexity of Iterated Reversible Computation

Eppstein

2023

TheoretiCS

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We study a class of functional problems reducible to computing $f^{(n)}(x)$ for inputs $n$ and $x$, where $f$ is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its definition, and in whether we require $f$ to have a polynomial-time inverse or to be computible by a reversible logic circuit. These problems are characterized by the complexity class $\mathsf{FP}^{\mathsf{PSPACE}}$, and include natural $\mathsf{FP}^{\mathsf{PSPACE}}$-complete problems in circuit complexity, cellular automata, graph algorithms, and the dynamical systems described by piecewise-linear transformations.

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Section: Related Workmentioning

confidence: 99%

The Complexity of Iterated Reversible Computation

Eppstein

2023

TheoretiCS

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“…A specific algorithm that has been frequently studied in this light is Thomasen's lollipop algorithm for finding a second Hamiltonian cycle in an (explicit) cubic graph by following a path in a much larger implicit graph defined from the given graph [42]. Although some inputs cause this algorithm to take exponential time [4,5,50] the complexity of finding a second Hamiltonian cycle in a different way is unknown, and was one of the motivating problems for the definition of PPA [35]. We formulate the same question in a different way, asking how hard it is to find the same cycle that Thomasen's algorithm finds, but again the complexity of this problem remains unknown.…”

Section: Related Workmentioning

confidence: 99%

“…This is an instance of the second leaf problem, in an implicit graph of maximum degree two (not necessarily a linear forest). It is known that some instances may cause Thomason's lollipop algorithm to take an exponential number of steps [4,5,26,50], but while this settles the complexity of this specific algorithm, it leaves open the complexity of the functional problem solved by the algorithm.…”

Section: Thomason's Lollipop Algorithmmentioning

confidence: 99%

The Complexity of Iterated Reversible Computation

Eppstein¹

2021

Preprint

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We define a complexity class IB as the class of functional problems reducible to computing f (n) (x) for inputs n and x, where f is a polynomial-time bijection. As we prove, the definition is robust against variations in the type of reduction used in its definition, and in whether we require f to have a polynomial-time inverse or to be computible by a reversible logic circuit. We relate IB to other standard complexity classes, and demonstrate its applicability by finding natural IB-complete problems in circuit complexity, cellular automata, graph algorithms, and the dynamical systems described by piecewise-linear transformations.

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A short note on graphs with long Thomason chains

Cited by 2 publications

References 3 publications

The Complexity of Iterated Reversible Computation

The Complexity of Iterated Reversible Computation

The Complexity of Iterated Reversible Computation

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