Combinatorial Bounds on Nonnegative Rank and Extended Formulations

An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes. The main known lower bounds on the minimum sizes of extended formulations for fixed polytope P (Yannakakis 1991) are closely related to the concept of nondeterministic communication complexity. We study the relative power and limitations of the bounds on several examples.

💡 Research Summary

The paper investigates the relationship between the extension complexity of a polytope P and the nonnegative rank of its slack matrix S(P), focusing on the combinatorial tool of rectangle coverings. An extended formulation of P is a higher‑dimensional polytope Q together with a linear projection that maps Q onto P; the size of the formulation is the number of facets of Q. Yannakakis (1991) showed that the minimum size of any extended formulation, denoted xc(P), equals the nonnegative rank of S(P). The nonnegative rank is the smallest integer r for which S can be written as U Vᵀ with nonnegative matrices U∈ℝ^{m×r} and V∈ℝ^{n×r}. This factorisation is equivalent to covering the support of S (the set of its non‑zero entries) by r combinatorial rectangles, i.e., Cartesian products of subsets of rows and columns. Consequently, the rectangle‑covering number rc(S) provides a lower bound on the nonnegative rank and thus on xc(P).

The authors first give a geometric reinterpretation of Yannakakis’s result. They introduce the notion of a slack generating set—a collection of nonnegative vectors that can express every slack vector of P as a nonnegative combination. They prove that the size of a smallest slack generating set coincides with the nonnegative rank of S(P) and therefore with xc(P). This viewpoint connects the lattice embedding of face lattices (the “meet‑faithful” embedding) with the algebraic factorisation.

Next, the paper studies both upper and lower bounds on rc(S). For upper bounds, they prove that if a polytope has n vertices and each facet contains at most k vertices, then
 rc(S) ≤ O(k² log n).
This improves over the trivial logarithmic bound and shows that many polytopes with sparse facet structure admit relatively small rectangle coverings.

For lower bounds, the authors reinterpret classic combinatorial techniques in the language of rectangle graphs. The rectangle graph of a nonnegative matrix has a vertex for each non‑zero entry and edges joining entries that lie in a common rectangle. Its chromatic number equals rc(S), while its clique number corresponds to the well‑known fooling‑set bound of Dietzfelbinger et al. (1996). Using this, they show that for the d‑dimensional cube and for the Birkhoff polytope the rectangle covering number equals the number of facets; consequently, their extension complexities are exactly the facet counts.

They also analyze neighborly polytopes. For a d‑dimensional neighborly polytope with Ω(d²) vertices they prove rc(S)=Ω(d²), implying that any extended formulation must have at least quadratic size in the dimension.

Importantly, the authors identify a limitation of the rectangle‑covering approach: for any d‑dimensional polytope the fooling‑set (clique) bound cannot exceed (d + 1)², which matches the best known combinatorial lower bound. Hence, purely combinatorial rectangle coverings cannot yield stronger lower bounds than this quadratic barrier.

The paper further connects rectangle coverings to communication complexity. The deterministic communication complexity of the support matrix equals ⌈log₂ rc(S)⌉, while recent results on nondeterministic communication complexity (Huang & Sudakov, 2020) provide additional insight into when rectangle coverings are tight or loose.

Overall, the work offers a comprehensive treatment of the combinatorial rectangle‑covering method as a tool for bounding extension complexity. It supplies new upper bounds for sparse‑facet polytopes, establishes tight lower bounds for the cube, Birkhoff polytope, and neighborly polytopes, and clarifies the intrinsic limitations of this approach. By bridging geometric, algebraic, graph‑theoretic, and communication‑complexity perspectives, the paper both consolidates existing knowledge and points toward the need for techniques beyond rectangle coverings to achieve stronger lower bounds on extended formulations.