A Formal Disproof of Hirsch Conjecture - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2023

A Formal Disproof of Hirsch Conjecture

Xavier Allamigeon
Quentin Canu
  • Function : Author
Pierre-Yves Strub
  • Function : Author

Abstract

The purpose of this paper is the formal verification of a counterexample of Santos et al. to the so-called Hirsch Conjecture on the diameter of polytopes (bounded convex polyhedra). In contrast with the pen-and-paper proof, our approach is entirely computational: we have implemented in Coq and proved correct an algorithm that explicitly computes, within the proof assistant, vertex-edge graphs of polytopes as well as their diameter. The originality of this certificate-based algorithm is to achieve a tradeoff between simplicity and efficiency. Simplicity is crucial in obtaining the proof of correctness of the algorithm. This proof splits into the correctness of an abstract algorithm stated over proof-oriented data types and the correspondence with a low-level implementation over computation-oriented data types. A special effort has been made to reduce the algorithm to a small sequence of elementary operations (e.g. matrix multiplications, basic routines on graphs), in order to make the derivation of the correctness of the low-level implementation more transparent. Efficiency allows us to scale up to polytopes with a challenging combinatorics. For instance, we formally check the two counterexamples to Hirsch conjecture due to Matschke, Santos and Weibel, respectively 20- and 23-dimensional polytopes with 36425 and 73224 vertices involving rational coefficients with up to 20 digits. We also illustrate the performance of the method by computing the list of vertices or the diameter of well-known classes of polytopes, such as (polars of) cyclic polytopes involved in McMullen's Upper Bound Theorem.

Dates and versions

hal-04368266 , version 1 (31-12-2023)

Licence

Identifiers

Cite

Xavier Allamigeon, Quentin Canu, Pierre-Yves Strub. A Formal Disproof of Hirsch Conjecture. CPP 2023 - 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, Jan 2023, Boston, United States. pp.17-29, ⟨10.1145/3573105.3575678⟩. ⟨hal-04368266⟩
27 View
0 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More