Abstract
AbstractThe objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. Spanning trees and perfect matchings are simple graph-theoretic objects, and the difference between them has deep mathematical roots. A matrix's determinant is the number of spanning trees while its permanent is the number of perfect matchings. Counting is closely associated with sampling. This chapter explores how to generate random matchings, and hence count them approximately, using a Markov chain that mixes in polynomial time. It considers the special case of planar graphs, such as the square lattice, to demonstrate that the number of perfect matchings is in P. It also discusses the implications of this fact for statistical physics and looks at how to find exact solutions for many physical models in two dimensions, including the Ising model.
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.
