The Computational Complexity of the Tutte Plane: the Bipartite Case

1992 ◽  
Vol 1 (2) ◽  
pp. 181-187 ◽  
Author(s):  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Ising Model ◽  
Partition Function ◽  
Polynomial Time ◽  
Spanning Trees ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Number Of Spanning Trees ◽  
Bipartite Case ◽  
Complete Characterisation

Along different curves and at different points of the (x, y)-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P-hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

scholarly journals Complexity of Ising Polynomials

2012 ◽  
Vol 21 (5) ◽  
pp. 743-772 ◽  
Author(s):  
TOMER KOTEK
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
External Field ◽  
Polynomial Time ◽  
Tutte Polynomial ◽  
Study Phase ◽  
Exceptional Set ◽  
Exponential Time ◽  
Low Dimension ◽  
Special Case

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomialZ(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson.Z(G;x,y,z) generalizes a bivariate polynomialZ(G;t,y), which was studied in by Andrén and Markström.We consider the complexity ofZ(Gt,y) andZ(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show thatZ(G;x,y,z) is #P-hard to evaluate at all points in3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETHa dichotomy theorem stating that evaluations ofZ(G;t,y) either take exponential time in the number of vertices ofGto compute, or can be done in polynomial time. Finally, we give an algorithm for computingZ(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

On the computational complexity of the Jones and Tutte polynomials

1990 ◽  
Vol 108 (1) ◽  
pp. 35-53 ◽  
Author(s):  
F. Jaeger ◽  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Tutte Polynomials ◽  
Special Points ◽  
Alternating Link ◽  
Special Case

AbstractWe show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, y)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y)-plane is #P-hard except when (x − 1)(y − 1) = 1 or when (x, y) equals (1, 1), (−1, −1), (0, −1), (−1, 0), (i, −i), (−i, i), (j, j2), (j2, j) where j = e2πi/3

Knots, matroids and the Ising model

1993 ◽  
Vol 113 (1) ◽  
pp. 107-139 ◽  
Author(s):  
W. Schwärzler ◽  
D. J. A. Welsh
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Knot Theory ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Kauffman Bracket ◽  
Link Diagrams ◽  
Bracket Polynomial

AbstractA polynomial is defined on signed matroids which contains as specializations the Kauffman bracket polynomial of knot theory, the Tutte polynomial of a matroid, the partition function of the anisotropic Ising model, the Kauffman–Murasugi polynomials of signed graphs. It leads to generalizations of a theorem of Lickorish and Thistlethwaite showing that adequate link diagrams do not represent the unknot. We also investigate semi-adequacy and the span of the bracket polynomial in this wider context.

scholarly journals Distance Hereditary Graphs and the Interlace Polynomial

2007 ◽  
Vol 16 (6) ◽  
pp. 947-973 ◽  
Author(s):  
JOANNA A. ELLIS-MONAGHAN ◽  
IRASEMA SARMIENTO
Keyword(s):  
Computational Complexity ◽  
Dna Sequencing ◽  
Planar Graph ◽  
Polynomial Time ◽  
Tutte Polynomial ◽  
Circle Graph ◽  
Open Questions ◽  
The One ◽  
Parallel Graph

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].We define the γ invariant as the coefficient of x1in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1whenntrue twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

scholarly journals Critical temperatures of the Ising model on Sierpiñski fractal lattices

2020 ◽  
Vol 244 ◽  
pp. 01013
Author(s):  
Michel Perreau
Keyword(s):  
Fractal Dimension ◽  
Critical Temperature ◽  
Ising Model ◽  
Partition Function ◽  
Fractal Dimensions ◽  
Simple Expression ◽  
Square Lattice ◽  
Critical Temperatures ◽  
Wide Range ◽  
Sierpinski Carpets

We report our latest results of the spectra and critical temperatures of the partition function of the Ising model on deterministic Sierpiñski carpets in a wide range of fractal dimensions. Several examples of spectra are given. When the fractal dimension increases (and correlatively the lacunarity decreases), the spectra aggregates more and more tightly along the spectrum of the regular square lattice. The single real root vc, comprised between 0 and 1, of the partition function, which corresponds to the critical temperature Tc through the formula vc = tanh(1/Tc), reliably fits a power law of exponent k where k is the segmentation step of the fractal structure. This simple expression allows to extrapolate the critical temperature for k → ∞. The plot of the logarithm of this extrapolated critical temperature versus the fractal dimension appears to be reliably linear in a wide range of fractal dimensions, except for highly lacunary structures of fractal dimensions close from 1 (the dimension of a quasilinear lattice) where the critical temperature goes to 0 and its logarithm to −∞.

scholarly journals Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

2021 ◽  
pp. 1-17
Author(s):  
Martin Dyer ◽  
Marc Heinrich ◽  
Mark Jerrum ◽  
Haiko Müller
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Polynomial Time ◽  
Line Graph ◽  
Glauber Dynamics ◽  
Monte Carlo Algorithm ◽  
Line Graphs ◽  
Time Approximation ◽  
On Line ◽  
Antiferromagnetic Ising Model

Abstract We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514–527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.

On the Complexity of Deadlock Recovery

1986 ◽  
Vol 9 (3) ◽  
pp. 323-342
Author(s):  
Joseph Y.-T. Leung ◽  
Burkhard Monien
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Arbitrary Number ◽  
The Other ◽  
Np Hard ◽  
Total Cost ◽  
Other Hand ◽  
Input Parameters ◽  
Resource Type

We consider the computational complexity of finding an optimal deadlock recovery. It is known that for an arbitrary number of resource types the problem is NP-hard even when the total cost of deadlocked jobs and the total number of resource units are “small” relative to the number of deadlocked jobs. It is also known that for one resource type the problem is NP-hard when the total cost of deadlocked jobs and the total number of resource units are “large” relative to the number of deadlocked jobs. In this paper we show that for one resource type the problem is solvable in polynomial time when the total cost of deadlocked jobs or the total number of resource units is “small” relative to the number of deadlocked jobs. For fixed m ⩾ 2 resource types, we show that the problem is solvable in polynomial time when the total number of resource units is “small” relative to the number of deadlocked jobs. On the other hand, when the total number of resource units is “large”, the problem becomes NP-hard even when the total cost of deadlocked jobs is “small” relative to the number of deadlocked jobs. The results in the paper, together with previous known ones, give a complete delineation of the complexity of this problem under various assumptions of the input parameters.

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