  Goemans and Rothvoss (SODA'14) gave a framework for solving problems in time
$enc(P)^{2^{O(N)}}enc(Q)^{O(1)}$ that can be described as finding a point in
$\text{int.cone}(P\cap\mathbb{Z}^N)\cap Q$, where $P,Q\subset\mathbb{R}^N$ are
(bounded) polyhedra. This framework can be used to solve various scheduling
problems, but the encoding length $enc(P)$ usually involves large parameters
like the makespan. We describe three tools to improve the framework by Goemans
and Rothvoss: Problem-specific preprocessing, LP relaxation techniques and a
new bound for the number of vertices of the integer hull.
  In particular, applied to the classical scheduling problem $P||C_{\max}$,
these tools each improve the running time from $(\log(C_{\max}))^{2^{O(d)}}
enc(I)^{O(1)}$ to the possibly much better
$(\log(p_{\max}))^{2^{O(d)}}enc(I)^{O(1)}$. Here, $p_{\max}$ is the largest
processing time, $d$ is the number of different processing times, $C_{\max}$ is
the makespan and $enc(I)$ is the encoding length of the instance. This running
time is FPT w.r.t. parameter $d$ if $p_{\max}$ is given in unary. We obtain
similar results for various other problems. Moreover, we show how a balancing
result by Govzmann et al. can be used to speed up an additive approximation
scheme by Buchem et al. (ICALP'21) in the high-multiplicity setting.
  On the complexity side, we use reductions from the literature to provide new
parameterized lower bounds for $P||C_{\max}$ and to show that the improved
running time of the additive approximation algorithm is probably optimal.
Finally, we show that the big open question asked by Mnich and van Bevern
(Comput. Oper. Res. '18) whether $P||C_{\max}$ is FPT w.r.t. the number of job
types $d$ has the same answer as the question whether $Q||C_{\max}$ is FPT
w.r.t. the number of job and machine types $d+\tau$ (all in high-multiplicity
encoding). The same holds for objective $C_{\min}$.
