Standard mixed-integer programming formulations for the stable set problem on $n$-node graphs require $n$ integer variables. We prove that this is almost optimal: We give a family of $n$-node graphs for which every polynomial-size MIP formulation requires $\Omega(n/\log^2 n)$ integer variables. By a polyhedral reduction we obtain an analogous result for $n$-item knapsack problems. In both cases, this improves the previously known bounds of $\Omega(\sqrt{n}/\log n)$ by Cevallos, Weltge & Zenklusen (SODA 2018). To this end, we show that there exists a family of $n$-node graphs whose stable set polytopes satisfy the following: any $(1+\varepsilon/n)$-approximate extended formulation for these polytopes, for some constant $\varepsilon > 0$, has size $2^{\Omega(n/\log n)}$. Our proof extends and simplifies the information-theoretic methods due to G\"o\"os, Jain & Watson (FOCS 2016, SIAM J. Comput. 2018) who showed the same result for the case of exact extended formulations (i.e. $\varepsilon = 0$).

翻译：标准混合整数规划公式在$n$节点图的稳定集问题中需要$n$个整数变量。我们证明这几乎是最优的：构造了一族$n$节点图，使得任意多项式大小的混合整数规划公式需要$\Omega(n/\log^2 n)$个整数变量。通过多面体约简，我们得到了$n$项背包问题的类似结论。在这两种情形下，该结果改进了Cevallos、Weltge与Zenklusen（SODA 2018）先前证明的$\Omega(\sqrt{n}/\log n)$下界。为此，我们展示存在一族$n$节点图，其稳定集多面体满足：对于某个常数$\varepsilon > 0$，这些多面体的任意$(1+\varepsilon/n)$-近似扩展公式的规模为$2^{\Omega(n/\log n)}$。我们的证明扩展并简化了Göös、Jain与Watson（FOCS 2016，SIAM J. Comput. 2018）在精确扩展公式情形（即$\varepsilon = 0$）中所用的信息论方法。