  A $(\phi,\epsilon)$-expander-decomposition of a graph $G$ (with $n$ vertices
and $m$ edges) is a partition of $V$ into clusters $V_1,\ldots,V_k$ with
conductance $\Phi(G[V_i]) \ge \phi$, such that there are at most $\epsilon m$
inter-cluster edges. Such a decomposition plays a crucial role in many graph
algorithms. We give a randomized $\tilde{O}(m/\phi)$ time algorithm for
computing a $(\phi, \phi\log^2 {n})$-expander decomposition. This improves upon
the $(\phi, \phi\log^3 {n})$-expander decomposition also obtained in
$\tilde{O}(m/\phi)$ time by [Saranurak and Wang, SODA 2019] (SW) and brings the
number of inter-cluster edges within logarithmic factor of optimal.
  One crucial component of SW's algorithm is non-stop version of the
cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut
player does not stop when it gets from the matching player an unbalanced sparse
cut, but continues to play on a trimmed part of the large side. The crux of our
improvement is the design of a non-stop version of the cleverer cut player of
[Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of
OSSV uses a more sophisticated random walk, a subtle potential function, and
spectral arguments. Designing and analysing a non-stop version of this game was
an explicit open question asked by SW.
