Complexity: Knots, Colourings and Countings

Welsh, Dominic

doi:10.1017/cbo9780511752506

Cited by 295 publications

(319 citation statements)

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“…The Tutte polynomial is known as a basic invariant of a graph structure. For a given graph G = (V, E), the Tutte polynomial is defined as [41]:…”

Section: Frontier-based Methods For Various Graph Problemsmentioning

confidence: 99%

Techniques of BDD/ZDD: Brief History and Recent Activity

Minato

2013

IEICE Trans. Inf. & Syst.

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SUMMARYDiscrete structures are foundational material for computer science and mathematics, which are related to set theory, symbolic logic, inductive proof, graph theory, combinatorics, probability theory, etc. Many problems solved by computers can be decomposed into discrete structures using simple primitive algebraic operations. It is very important to represent discrete structures compactly and to execute efficiently tasks such as equivalency/validity checking, analysis of models, and optimization. Recently, BDDs (Binary Decision Diagrams) and ZDDs (Zero-suppressed BDDs) have attracted a great deal of attention, because they efficiently represent and manipulate large-scale combinational logic data, which are the basic discrete structures in various fields of application. Although a quarter of a century has passed since Bryant's first idea, there are still a lot of interesting and exciting research topics related to BDD and ZDD. BDD/ZDD is based on in-memory data processing techniques, and it enjoys the advantage of using random access memory. Recent commodity PCs are equipped with gigabytes of main memory, and we can now solve large-scale problems which used to be impossible due to memory shortage. Thus, especially since 2000, the scope of BDD/ZDD methods has increased. This survey paper describes the history of, and recent research activity pertaining to, techniques related to BDD and ZDD.

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“…The Tutte polynomial is known as a basic invariant of a graph structure. For a given graph G = (V, E), the Tutte polynomial is defined as [41]:…”

Section: Frontier-based Methods For Various Graph Problemsmentioning

confidence: 99%

Techniques of BDD/ZDD: Brief History and Recent Activity

Minato

2013

IEICE Trans. Inf. & Syst.

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“…[27] it was further shown that there is an equivalence between classical partition functions with real couplings and quantum amplitudes for a certain certain class of quantum circuits known as Clifford circuits. When this mapping exists the graph underling the classical theory is planar with no magnetic fields and can be estimated with a polynomial time classical algorithm [28]. Also, deciding if a certain quantum circuit belongs to this equivalence class is classically easy.…”

Section: Computational Power Of Classical Modelsmentioning

confidence: 99%

Low depth quantum circuits for Ising models

et al. 2014

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A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond the present context. It holds for example for estimations of the Jones polynomial. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made "instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allows one to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetizations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.

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“…yFðG À eÞ if e is a loop; aFðG=eÞ þ bFðG À eÞ if e is an ordinary edge: The Tutte polynomial is an example of Tutte-Gröthendieck invariant for a ¼ b ¼ c ¼ 1 [26] and so, TðG; x; yÞ…”

Section: The Tutte Polynomialmentioning

confidence: 99%

“…Also, evaluating TðG; x; yÞ for specific points ðx; yÞ is NP-hard except for the points on the hyperbola ðx À 1Þðy À 1Þ¼ 1 or when ðx; yÞ equals ð1; 1Þ; ðÀ1; À1Þ; ð0; À1Þ; ðÀ1; 0Þ; ði; ÀiÞ; ðÀi; iÞ; ðj; j 2 Þ; ðj 2 ; jÞ where j ¼ e ð2pi=3Þ for which it can be done in polynomial time [16,26]. Vertigan [29] has extended this by showing that a similar result holds for planar graphs except that, in this case, for the additional points lying on the hyperbola ðx À 1Þðy À 1Þ¼ 2, the problem can be solved again in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

Computing the Tutte polynomial of Archimedean tilings

Garijo

Gegúndez

Márquez

et al. 2014

Applied Mathematics and Computation

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We describe an algorithm to compute the Tutte polynomial of large fragments of Archimedean tilings by squares, triangles, hexagons and combinations thereof. Our algorithm improves a well known method for computing the Tutte polynomial of square lattices. We also address the problem of obtaining Tutte polynomial evaluations from the symbolic expressions generated by our algorithm, improving the best known lower bound for the asymptotics of the number of spanning forests, and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice.

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Complexity: Knots, Colourings and Countings

Cited by 295 publications

References 0 publications

Techniques of BDD/ZDD: Brief History and Recent Activity

Techniques of BDD/ZDD: Brief History and Recent Activity

Low depth quantum circuits for Ising models

Computing the Tutte polynomial of Archimedean tilings

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