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

TL;DR

A polyhedral reduction is obtained for the stable set problem on jats:italic-node graphs by a polyhedral reduction and improves the previously known bounds of the jats:inline-formula integer variables.

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 $Ω(n/\log^2 n)$ 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 $Ω(\sqrt{n}/\log n)$ 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 $2^{Ω(n/\log n)}$. 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. $\varepsilon = 0$).

Figures (3)

• Figure 1: Construction of the graph $H$ (right), where $G$ is a path on three nodes (left) and $\mathcal{X}$ consists of only two colors (white and gray).
• Figure 2: Both pictures show the values of the gadget $g$ in white ($0$) and gray ($1$), where the columns and rows corresponds to the inputs in the order $(0,0,0), \, (0,0,1), \, (0,1,0)$, etc. The right picture shows a $0$-window $w = \prescript{}{[11]}{w}$ and the corresponding $1$-windows $\prescript{}{[00]}{w}$, $\prescript{}{[01]}{w}$, and $\prescript{}{[10]}{w}$.
• Figure 3: Illustration of a spanning subgraph of $\tilde{\mathfrak G}^1$, which shows that $\tilde{\mathfrak G}^1$ is connected. Note that every column and row has exactly $4$ gray entries, which means that every node in $\tilde{\mathfrak G}^1$ has degree $6$. Moreover, for every column, there is a column with inverted entries, which shows that $\tilde{\mathfrak G}^0$ and $\tilde{\mathfrak G}^1$ are isomorphic.

Theorems & Definitions (46)

• theorem 1
• theorem 2
• lemma 1
• proof
• theorem 3
• proposition 1
• theorem 4
• theorem 5
• proposition 2: CT06
• proposition 3