Abstract
A distribution is tractable if it is possible to approximately sample from the distribution in polynomial time. Here the ferromagnetic Ising model with unidrectional magnetic field is shown to be reducible to a standard distribution on matchings that is tractable. This provides an alternate method to the original Jerrum and Sinclair approach to show that the Ising distribution itself is tractable. Previous reductions of the Ising model to perfect matchings on different graphs exist, but these older distributions are not tractable. Also, the older reductions did not consider an external magnetic field, while the new reduction explictly includes such a field. The new reduction also helps to explain why the idea of canonical paths is so useful inapproximately sampling from both problems. In addition, the reduction allows any algorithm for matchings to immediately be applied to the Ising model. For instance, this immediately yields a fully polynomial time approximation scheme for the Ising model on a bounded degree graph with magnetization bounded away from 0, merely by invoking an existing algorithm for matchings.
Highlights
All problems in NP are reducible in polynomial time to an NP-complete problem, illustrating the difficulty of the NP-complete problem
A simulation reduction is an algorithm that takes a draw for one simulation problem, and uses it to construct a draw for a different simulation problem
In this work we present algorithms that reduce HTEIS to PMATCH and HTEIS to MATCH The HTEIS to MATCH reduction takes advantage of the unidirectional external magnetic field, and the weights associated with edges are polynomial in the weights assigned the original problem
Summary
All problems in NP are reducible in polynomial time to an NP-complete problem, illustrating the difficulty of the NP-complete problem. The main result of this paper states that the subgraphs version of the Ising model is simulation reducible to the simulation problem of drawing matchings in a graph in such a way as to give a new polynomial time algorithm for approximately sampling from the Ising model This links work of Jerrum and Sinclair on Markov chains for matchings [6] and the Ising model [7]. In this work we present algorithms that reduce HTEIS to PMATCH and HTEIS to MATCH The HTEIS to MATCH reduction takes advantage of the unidirectional external magnetic field, and the weights associated with edges are polynomial in the weights assigned the original problem. The reductions presented here give some idea of why conductance should work well for both the matchings problem and the Ising model, since the high temperature Ising expansion can be effectively viewed as a special instance of the matchings distribution.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
