  Mixed-integer linear programming (MILP) is at the core of many advanced
algorithms for solving fundamental problems in combinatorial optimization. The
complexity of solving MILPs directly correlates with their support size, which
is the minimum number of non-zero integer variables in an optimal solution. A
hallmark result by Eisenbrand and Shmonin (Oper. Res. Lett., 2006) shows that
any feasible integer linear program (ILP) has a solution with support size
$s\leq 2m\cdot\log(4m\Delta)$, where $m$ is the number of constraints, and
$\Delta$ is the largest coefficient in any constraint.
  Our main combinatorial result are improved support size bounds for ILPs.
  To improve granularity, we analyze for the largest $1$-norm $A_{\max}$ of any
column of the constraint matrix, instead of $\Delta$. We show a support size
upper bound of $s\leq m\cdot(\log(3A_{\max})+\sqrt{\log(A_{\max})})$, by
deriving a new bound on the -1 branch of the Lambert $\mathcal{W}$ function.
Additionally, we provide a lower bound of $m\log(A_{\max})$, proving our result
asymptotically optimal. Furthermore, we give support bounds of the form $s\leq
2m\cdot\log(1.46A_{\max})$. These improve upon the previously best constants by
Aliev. et. al. (SIAM J. Optim., 2018), because all our upper bounds hold
equally with $A_{\max}$ replaced by $\sqrt{m}\Delta$.
  Using our combinatorial result, we obtain the fastest known approximation
schemes (EPTAS) for the fundamental scheduling problem of makespan minimization
of uniformly related machines ($Q\mid\mid C_{\max}$).
