An algebraic approach to symmetric extended formulations

Extended formulations are an important tool to obtain small (even compact) formulations of polytopes by representing them as projections of higher dimensional ones. It is an important question whether a polytope admits a small extended formulation, i.e., one involving only a polynomial number of inequalities in its dimension. For the case of symmetric extended formulations (i.e., preserving the symmetries of the polytope) Yannakakis established a powerful technique to derive lower bounds and rule out small formulations. We rephrase the technique of Yannakakis in a group-theoretic framework. This provides a different perspective on symmetric extensions and considerably simplifies several lower bound constructions.

💡 Research Summary

The paper revisits Yannakakis’s seminal technique for proving lower bounds on the size of extended formulations (EFs) and places it within a group‑theoretic framework that explicitly handles symmetry. An extended formulation of a polytope P ⊂ ℝⁿ is a higher‑dimensional polytope Q ⊂ ℝᵈ together with a linear projection π such that π(Q)=P. The size of the EF is the number of inequalities defining Q; a “small” EF uses only polynomially many inequalities in n. Yannakakis showed that the nonnegative rank of the slack matrix S(P) provides a lower bound on the size of any EF: a small EF implies a low‑rank nonnegative factorisation of S(P).

When the polytope possesses a symmetry group G (e.g., the matching polytope, the traveling‑salesman polytope, or the cut polytope), one may demand that the EF respect this symmetry: G must act simultaneously on the vertices of Q, on the facets of Q, and on the projection map. Such “symmetric” EFs are considerably more restrictive, and Yannakakis’s original method required intricate combinatorial arguments to exploit these restrictions.

The authors’ main contribution is to replace those combinatorial arguments with a clean algebraic construction. They observe that the slack matrix S(P) is G‑invariant, i.e., g·S(P)=S(P) for all g∈G, when rows are indexed by vertices and columns by facets. By grouping rows and columns into G‑orbits, they define the orbit slack matrix Sᴼ(P). Crucially, the nonnegative rank of S(P) is at least the nonnegative rank of Sᴼ(P). Hence, any lower bound on the nonnegative rank of the much smaller orbit matrix automatically yields a lower bound for symmetric EFs.

To analyse Sᴼ(P) the paper brings in elementary representation theory. The space of real functions on the vertex set decomposes into G‑invariant subspaces (isotypic components). A G‑invariant nonnegative factorisation of S(P) must respect this decomposition, which forces the factor matrices to lie in the invariant subspace. Consequently, the dimension of the invariant subspace—called the invariant dimension—provides a concrete lower bound on the nonnegative rank of Sᴼ(P). In many classical examples this invariant dimension is linear or super‑linear in n, reproducing known lower bounds (e.g., Ω(n) for the perfect‑matching polytope) with a single line of algebra.

The authors also introduce the notion of a symmetric nonnegative factorisation: a factorisation S(P)=UVᵀ where both U and V are G‑equivariant. Such a factorisation is sufficient to construct a symmetric EF of size equal to the rank of the factorisation. Conversely, proving that no symmetric factorisation exists below a certain rank rules out symmetric EFs of that size. Using this criterion, they give streamlined proofs of existing lower bounds for the matching, cut, and traveling‑salesman polytopes, and they obtain new bounds for families of polytopes whose symmetry groups are non‑abelian or have multiple orbit types.

Beyond the theoretical results, the paper discusses algorithmic implications. The orbit‑matrix reduction suggests a practical preprocessing step: compute the orbit structure of G, form Sᴼ(P), and then apply standard nonnegative rank estimation techniques (e.g., semidefinite relaxations) on a dramatically smaller matrix. This could make symmetry‑aware EF lower‑bound computations feasible for instances that were previously out of reach.

In the concluding section the authors outline several promising directions. First, extending the framework to partial symmetries or to groups that act only on a subset of facets may capture many real‑world optimisation models. Second, developing efficient algorithms to decide the existence of low‑rank symmetric nonnegative factorizations could lead to automated EF design tools that respect problem symmetries. Third, exploring connections with communication complexity—where the nonnegative rank corresponds to deterministic protocols—might yield new insights into the inherent difficulty of symmetric optimisation problems.

Overall, the paper provides a unifying algebraic lens for symmetric extended formulations. By translating Yannakakis’s combinatorial arguments into the language of group actions and invariant subspaces, it not only simplifies existing lower‑bound proofs but also opens a systematic pathway for discovering new bounds and for designing symmetry‑aware optimisation algorithms.