  In this work, we develop a comparison procedure for the Modified log-Sobolev
Inequality (MLSI) constants of two reversible Markov chains on a finite state
space. Efficient comparison of the MLSI Dirichlet forms is a well known
obstacle in the theory of Markov chains. We approach this problem by
introducing a {\it regularized} MLSI constant which, under some assumptions,
has the same order of magnitude as the usual MLSI constant yet is amenable for
comparison and thus considerably simpler to estimate in certain cases. As an
application of this general comparison procedure, we provide a sharp estimate
of the MLSI constant of the switch chain on the the set of simple bipartite
regular graphs of size $n$ with a fixed degree $d$. Our estimate implies that
the total variation mixing time of the switch chain is of order $O_d(n\log n)$.
The result is optimal up to a multiple depending on $d$ and resolves a
long-standing open problem. We expect that the MLSI comparison technique
implemented in this paper will find further applications.
