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.

scholarly journals Some Applications of Wagner's Weighted Subgraph Counting Polynomial

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ferenc Bencs ◽  
Péter Csikvári ◽  
Guus Regts
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Line Graphs ◽  
Uniform Hypergraph ◽  
Edge Cover ◽  
Absolute Value ◽  
On Line ◽  
Ferromagnetic Ising Model ◽  
Cover Polynomial ◽  
Counting Polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.  

On Spanning and Dominating Circuits in Graphs

1977 ◽  
Vol 20 (2) ◽  
pp. 215-220 ◽  
Author(s):  
L. Lesniak-Foster ◽  
James E. Williamson
Keyword(s):  
Line Graph ◽  
Dominating Set ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Vertex Set ◽  
On Line ◽  
One To One

AbstractA set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

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.

scholarly journals Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

2016 ◽  
Vol 25 (4) ◽  
pp. 500-559 ◽  
Author(s):  
ANDREAS GALANIS ◽  
DANIEL ŠTEFANKOVIČ ◽  
ERIC VIGODA
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Maximum Degree ◽  
Hard Core ◽  
Regular Tree ◽  
Infinite Tree ◽  
Critical Activity ◽  
Antiferromagnetic Ising Model ◽  
Hard Core Model ◽  
Constant Maximum

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λc($\mathbb{T}_{\Delta}$) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented anFPTASfor the partition function when λ < λc($\mathbb{T}_{\Delta}$) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists εΔ> 0 such that (unless RP = NP) there is noFPRASfor approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λc($\mathbb{T}_{\Delta}$) < λ < λc($\mathbb{T}_{\Delta}$) + εΔ.We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding anFPTASfor the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is noFPRASfor approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree$\mathbb{T}_{\Delta}$. Our proof works by relating certain second moment calculations for random Δ-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

scholarly journals No matter how you mark the points on the fever curve – threatening shapes do not add to threat of climate change

2021 ◽  
Author(s):  
Mariam Katsarava ◽  
Helen Landmann ◽  
Robert Gaschler
Keyword(s):  
Climate Change ◽  
Line Graph ◽  
The Other ◽  
Line Graphs ◽  
Specific Data ◽  
Graph Comprehension ◽  
Content Domain ◽  
Subjective Assessments ◽  
On Line ◽  
Important Means

AbstractGraphs have become an increasingly important means of representing data, for instance, when communicating data on climate change. However, graph characteristics might significantly affect graph comprehension. The goal of the present work was to test whether the marking forms usually depicted on line-graphs, can have an impact on graph evaluation. As past work suggests that triangular forms might be related to threat, we compared the effect of triangular marking forms with other symbols (triangles, circles, squares, rhombi, and asterisks) on subjective assessments. Participants in Study 1 (N = 314) received 5 different line-graphs about climate change, each of them using one out of 5 marking forms. In Study 1, the threat and arousal ratings of the graphs with triangular marking shapes were not higher than those with the other marking symbols. Participants in Study 2 (N = 279) received the same graphs, yet without labels and indeed rated the graphs with triangle point markers as more threatening. Testing whether local rather than global spatial attention would lead to an impact of marker shape in climate graphs, Study 3 (N = 307) documented that a task demanding to process a specific data-point on the graph (rather than just the line graph as a whole) did not lead to an effect either. These results suggest that marking symbols can principally affect threat and arousal ratings but not in the context of climate change. Hence, in graphs on climate change, choice of point markers does not have to take potential side-effects on threat and arousal into account. These seem to be restricted to the processing of graphs where form aspects face less competition from the content domain on judgments.

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