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 Ω(n/log2n) 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
Ω(n/logn) 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+ε/n)-approximate
extended formulation for these polytopes, for some constant ε>0,
has size 2Ω(n/logn). 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. ε=0).Comment: 35 page