1 research outputs found
Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack
Standard mixed-integer programming formulations for the stable set problem on
-node graphs require integer variables. We prove that this is almost
optimal: We give a family of -node graphs for which every polynomial-size
MIP formulation requires integer variables. By a
polyhedral reduction we obtain an analogous result for -item knapsack
problems. In both cases, this improves the previously known bounds of
by Cevallos, Weltge & Zenklusen (SODA 2018).
To this end, we show that there exists a family of -node graphs whose
stable set polytopes satisfy the following: any -approximate
extended formulation for these polytopes, for some constant ,
has size . 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. ).Comment: 35 page