Formally Certifying the Vertex Set of a Polyhedron Faster than Informal Enumeration

2026-10-08

AI Summary

This study addresses the high computational cost of verifying polytope vertex enumerations and the inefficiency of traditional formal methods. We propose a certificate-based formal verification framework that introduces an abstract simplicial complex generalizing normal fan triangulations as a novel completeness criterion. This approach reduces expensive numerical computations to membership tests and inexpensive combinatorial checks, substantially lowering verification complexity. The framework is implemented using the Rocq proof assistant with OCaml extraction. Experimental evaluations demonstrate certified speedups of 1.5× to 5× over lrslib across various polytopes, while rigorously establishing formal correctness guarantees.

Abstract

The computation of the vertices of a polyhedron described by a system of linear inequalities is a central problem in polyhedral computation. It is a fundamental step in the conversion between H-representations, by linear inequalities, and V-representations, by vertices and extreme rays. This operation plays an important role both in the study of polyhedra and their combinatorics in mathematics and in applications to software and system verification. We present a certificate-based approach for formally verifying the computation of the vertices of a polyhedron. Given an informally computed list of vertices, our method allows to certify in the proof assistant Rocq that the list is complete, or even exact. The cornerstone of the method is a new completeness criterion based on an abstract simplicial complex that generalizes a triangulation of the normal fan of the polyhedron. A significant advantage over previous approaches is that the usually expensive numerical computations are essentially reduced to membership tests to the polyhedron, while the other steps are cheap combinatorial tests. We implement the certification method and prove its correctness in the proof assistant Rocq. We experiment with it on a variety of polyhedra, including Birkhoff polytopes, cross-polytopes, cubes, permutahedra, hypersimplices, and high-dimensional polytopes involved in the disproof of the Hirsch conjecture. Our experiments show that certification with the Rocq-to-OCaml extracted checker is typically 1.5x to over 5x faster than vertex enumeration by the state-of-the-art informal C implementation lrslib of the reverse search method.

Problem

• polyhedron vertex computation
• formal verification
• H-representation to V-representation
• polyhedral computation
• certificate-based certification

Innovation

• Formal Verification
• Polyhedron Vertices
• Abstract Simplicial Complex
• Certificate-based Approach
• Proof Assistant