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 ($Q\mid\mid C_{\max}$).

翻译：混合整数线性规划（MILP）是许多解决组合优化基本问题的高级算法的核心。求解MILP的复杂度与其支撑规模直接相关，支撑规模是指最优解中非零整数变量的最小数量。Eisenbrand和Shmonin（Oper. Res. Lett., 2006）的标志性结果表明，任何可行的整数线性规划（ILP）都存在一个支撑规模为$s\leq 2m\cdot\log(4m\Delta)$的解，其中$m$是约束数量，$\Delta$是任意约束中的最大系数。我们的主要组合结果是改进了ILP的支撑规模界。为了提高粒度，我们分析了约束矩阵中任意列的最大1-范数$A_{\max}$，而非$\Delta$。通过推导Lambert $\mathcal{W}$函数-1分支的新边界，我们得到了支撑规模上界$s\leq m\cdot(\log(3A_{\max})+\sqrt{\log(A_{\max})})$。此外，我们提供了下界$m\log(A_{\max})$，证明了我们的结果在渐近意义下是最优的。进一步地，我们给出了形式为$s\leq 2m\cdot\log(1.46A_{\max})$的支撑界。这些结果改进了Aliev等人（SIAM J. Optim., 2018）此前的最佳常数，因为我们的所有上界在将$A_{\max}$替换为$\sqrt{m}\Delta$时同样成立。利用我们的组合结果，我们获得了均匀相关机器上最大完工时间最小化（$Q\mid\mid C_{\max}$）这一基本调度问题的最快已知近似方案（EPTAS）。