Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack
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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/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 Ω(√(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+ε/n)-approximate extended formulation for these polytopes, for some constant ε > 0, has size 2^Ω(n/log n). Our proof extends and simplifies the information-theoretic methods due to Göös, Jain Watson (FOCS 2016, SIAM J. Comput. 2018) who showed the same result for the case of exact extended formulations (i.e. ε = 0).
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