Two Linear Programs are Better than One
The extension complexity $\operatorname{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that projects onto it. Motivated by the complexity of neural networks, Hertrich and Loho introduced the virtual extension complexity $\operatorname{vxc}(P)$ of a polytope $P$, which is the minimum of $\operatorname{xc}(Q) + \operatorname{xc}(R)$ among all polytopes $Q$ and $R$ such that $P+Q=R$. We show the following results pertaining to virtual extension complexity. First, we resolve the core open problem on the topic by showing the virtual extension complexity can be exponentially smaller than extension complexity. Second, we show there are matroid base polytopes with exponen...
| Authors | Alexander E. Black, Christoph Hertrich, Stefan Kober et al. |
|---|---|
| Primary category | math.CO |
| Published | 11 October 2026 |
| Last updated at source | 11 October 2026 |
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