Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

💡 Research Summary

The paper resolves a long‑standing open problem posed by Yannakakis in 1991: it proves that no polynomial‑size linear program (LP), symmetric or not, can exactly represent the traveling salesman polytope (TSP), the cut polytope, or the stable‑set polytope. The authors achieve this by establishing exponential lower bounds on the extension complexity of these polytopes, using a novel bridge between one‑way quantum communication protocols and semidefinite programming (SDP) reformulations of LPs.

The core technical framework builds on Yannakakis’s factorization theorem, which equates the extension complexity of a polytope P with the nonnegative rank of its slack matrix S. The authors generalize this to conic extensions, showing that the SDP extension complexity equals the PSD rank of S. They then define a specific 2ⁿ × 2ⁿ matrix M(n) whose support matrix encodes the classic Disjointness problem. By invoking Razborov’s Ω(n) lower bound on the nondeterministic communication complexity of Disjointness (as formalized by de Wolf), they prove that the nonnegative rank of M is 2^{Ω(n)}. Consequently, any extended formulation of the cut polytope CUT(n) must contain an exponential number of inequalities.

Through linear reductions, the same exponential lower bound propagates to the stable‑set polytope STAB(G) for an infinite family of graphs (yielding a 2^{Ω(√n)} lower bound) and to the TSP polytope TSP(n) (also 2^{Ω(√n)}). Thus, the previously known Ω(n) lower bound for symmetric extensions is superseded by unconditional, super‑polynomial bounds that hold without any symmetry assumptions.

A second major contribution is the establishment of a tight correspondence between SDP extensions and one‑way quantum communication. The authors show that a rank‑r PSD factorization of a nonnegative matrix M yields a quantum protocol using log r + O(1) qubits that computes M in expectation; conversely, any q‑qubit one‑way quantum protocol for M induces a PSD factorization of rank at most 2^{O(q)}. This bidirectional translation leads to several corollaries: (i) any d‑dimensional 0/1‑slack polytope admits an SDP extension of size O(d); (ii) there exists an exponential separation between PSD rank and nonnegative rank, demonstrated by the matrix M(n), whose PSD rank is O(n) while its nonnegative rank is 2^{Ω(n)}.

The paper’s methodology mirrors recent “quantum‑inspired” lower‑bound techniques, yet it provides fully explicit polytopes and elementary proofs. By first constructing an efficient quantum protocol (or equivalently a low‑rank PSD factorization) for M(n) and then leveraging classical communication lower bounds, the authors invert the usual direction of argument. This approach not only settles Yannakakis’s open question but also deepens the interplay between combinatorial optimization, extended formulations, and quantum information theory.

Overall, the work demonstrates that polynomial‑size LPs cannot capture the exact combinatorial structure of TSP, cut, or stable‑set problems, and it introduces a powerful quantum‑communication lens for analyzing extension complexity, opening new avenues for both lower‑bound research and the design of compact SDP relaxations.