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：我们构造了一族$n$节点图，使得任何多项式规模的混合整数规划公式都需要$\Omega(n/\log^2 n)$个整数变量。通过多面体约化，我们为$n\)项背包问题获得了一个类似的结果。在这两种情形中，这改进了Cevallos、Weltge和Zenklusen（SODA 2018）先前已知的$\Omega(\sqrt{n}/\log n)$下界。为此，我们证明存在一族$n\)节点图，其稳定集多面体满足以下性质：对于某个常数$\varepsilon > 0\)，这些多面体的任何$(1+\varepsilon/n)$-近似扩展公式的大小均为$2^{\Omega(n/\log n)}$。我们的证明扩展并简化了Göös、Jain和Watson（FOCS 2016，SIAM J. Comput. 2018）的信息论方法，他们证明了精确扩展公式（即$\varepsilon = 0$）情况下的相同结果。