A counterexample to the Hirsch conjecture (1006.2814v3)

Abstract: The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional polytope with $n$ facets cannot have (combinatorial) diameter greater than $n-d$. That is, that any two vertices of the polytope can be connected by a path of at most $n-d$ edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets which violates a certain generalization of the $d$-step conjecture of Klee and Walkup.

• The paper constructs the first counterexample to the Hirsch Conjecture, a 43-dimensional polytope with 86 facets whose diameter exceeds the proposed bound.
• The paper's methodology shows how constructions like the one-point suspension can build higher-dimensional polytopes whose diameter exceeds the Hirsch boundary.
• The findings require re-evaluating algorithms dependent on the conjecture, such as the simplex method, and point to new research directions in polytope theory.

A Counterexample to the Hirsch Conjecture

In the domain of polytope theory, the Hirsch Conjecture, posited in 1957 by Warren M. Hirsch, made a significant assertion regarding the combinatorial properties of polytopes. It conjectured that the graph of a $d$-dimensional polytope with $n$ facets cannot possess a combinatorial diameter greater than $n-d$. This proposition held immense implications for the computational geometry and operations research communities, particularly in the context of the efficiency of the simplex method in linear programming.

The paper by Francisco Santos presents the first known counterexample to the Hirsch Conjecture, thereby resolving a long-standing question in geometric combinatorics. The discussed polytope has a dimension of 43 and encompasses 86 facets, yet it demonstrates a combinatorial diameter that exceeds the Hirsch bound by one. This counterexample fundamentally alters the landscape of polytope theory, implying that the conjecture does not universally hold.

Main Contributions

1. Construction of a Non-Hirsch Polytope: The author constructs a 43-dimensional polytope with 86 facets, confirming a diameter of at least 44. This refutation is based on extending a 5-dimensional polytope, exhibiting a core violation of a generalized $d$-step conjecture relevant to polytopes.
2. Methodological Framework: The paper employs a strong $d$-step theorem framework for spindles (a polytope with two distinguished vertices that are as distant as possible under facet constraints). A key result demonstrates the feasibility of constructing higher-dimensional polytopes where the diameter exceeds the Hirsch boundary by employing operations like the one-point-suspension.
3. Asymptotic Violation: Through successive constructions (by taking products and glueing polytopes), the paper deduces the existence of an infinite family of polytopes of fixed dimension whose diameters grow super-linearly with the number of facets. This demonstrates that without the Hirsch constraint, polytope diameter can surpass even polynomial bounds, assuming a linear increase with facets.

Implications and Future Directions

The consequences of Santos' work are multifaceted. Practically, this counterexample suggests a reevaluation of algorithms relying on the Hirsch conjecture as a theoretical underpinning. Specifically, the simplex method, while efficacious empirically, cannot be explained solely as a consequence of the conjecture. This insight directs future investigations towards either elucidating the typical efficiency of the simplex method in practice or pursuing alternative theoretical justifications for its performance.

Theoretically, while the paper addresses the Hirsch conjecture's inapplicability, it opens avenues for new conjectures or adjusted bounds. For instance, variants such as the polynomial Hirsch conjecture or other forms of bounded diameter calculations now serve as pertinent research directions.

Additionally, the combinatorial techniques and constructions developed provide a rich toolkit for exploring analogous problems in higher-dimensional geometry and related fields. Future work could explore bounds or characteristics of polytopes in restricted settings, perhaps under additional constraints such as symmetry or specific vertex attributes.

In conclusion, Santos' paper is a significant contribution to polytope theory and computational complexity, reframing our understanding of polytope diameters in high-dimensional spaces. The presented work not only resolves a historic conjecture but also guides the community towards exploring novel hypotheses and methodologies in geometric and combinatorial optimization.