Phase transitions in the complexity of counting

Galanis, Andreas

Title:

Phase transitions in the complexity of counting

dc.contributor.advisor	Vigoda, Eric
dc.contributor.author	Galanis, Andreas
dc.contributor.committeeMember	Perkins, Will
dc.contributor.committeeMember	Randall, Dana
dc.contributor.committeeMember	Stefankovic, Daniel
dc.contributor.committeeMember	Tetali, Prasad
dc.contributor.committeeMember	Vempala, Santosh
dc.contributor.department	Computer Science
dc.date.accessioned	2014-08-27T13:33:27Z
dc.date.available	2014-08-27T13:33:27Z
dc.date.created	2014-08
dc.date.issued	2014-05-16
dc.date.submitted	August 2014
dc.date.updated	2014-08-27T13:33:27Z
dc.description.abstract	A recent line of works established a remarkable connection for antiferromagnetic 2-spin systems, including the Ising and hard-core models, showing that the computational complexity of approximating the partition function for graphs with maximum degree \Delta undergoes a computational transition that coincides with the statistical physics uniqueness/non-uniqueness phase transition on the infinite \Delta-regular tree. Despite this clear picture for 2-spin systems, there is little known for multi-spin systems. We present the first analog of the above inapproximability results for multi-spin systems. The main difficulty in previous inapproximability results was analyzing the behavior of the model on random \Delta-regular bipartite graphs, which served as the gadget in the reduction. To this end one needs to understand the moments of the partition function. Our key contribution is connecting: (i) induced matrix norms, (ii) maxima of the expectation of the partition function, and (iii) attractive fixed points of the associated tree recursions (belief propagation). We thus obtain a generic analysis of the Gibbs distribution of any multi-spin system on random regular bipartite graphs. We also treat in depth the k-colorings and the q-state antiferromagnetic Potts models. Based on these findings, we prove that for \Delta constant and even k<\Delta, it is NP-hard to approximate within an exponential factor the number of k-colorings on triangle-free \Delta-regular graphs. We also prove an analogous statement for the antiferromagnetic Potts model. Our hardness results for these models complement the conjectured regime where the models are believed to have efficient approximation schemes. We systematize the approach to obtain a general theorem for the computational hardness of counting in antiferromagnetic spin systems, which we ultimately use to obtain the inapproximability results for the k-colorings and q-state antiferromagnetic Potts models, as well as (the previously known results for) antiferromagnetic 2-spin systems. The criterion captures in an appropriate way the statistical physics uniqueness phase transition on the tree.
dc.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/52211
dc.language.iso	en_US
dc.publisher	Georgia Institute of Technology
dc.subject	Phase transitions
dc.subject	Partition function
dc.subject	Approximation algorithms
dc.subject	Spin systems
dc.title	Phase transitions in the complexity of counting
dc.type	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.corporatename	College of Computing
relation.isOrgUnitOfPublication	c8892b3c-8db6-4b7b-a33a-1b67f7db2021
thesis.degree.level	Doctoral