Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack

TL;DR

This paper establishes near-optimal lower bounds on the size of mixed-integer programming formulations for stable set and knapsack problems, demonstrating that polynomial-size formulations require a super-polynomial number of integer variables.

Contribution

It proves that for certain graphs and knapsack instances, any polynomial-size MIP formulation must have a super-logarithmic number of integer variables, improving previous bounds.

Findings

Polynomial-size MIP formulations require (n/log^2 n) integer variables for some graphs.

Any (/n) approximate extended formulation for certain stable set polytopes has exponential size.

The proof extends information-theoretic methods to approximate formulations, simplifying prior approaches.

Abstract

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. $\varepsilon = 0$).