The width of five-dimensional prismatoids

Matschke, Benjamin; Santos, Francisco; Weibel, Christophe

doi:10.1112/plms/pdu064

Cited by 27 publications

(47 citation statements)

References 11 publications

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“…In this paper, we present the first steps of a formalization of the theory of convex polyhedra in the proof assistant Coq. A motivation for using Coq comes from the longer term objective of formally proving some mathematical results relying on large-scale computation (e.g., the counterexample to the Hirsch conjecture given in [19], a polytope in dimension 20 with 36 425 vertices). The originality of our approach lies in the fact that our formalization is carried out in an effective way, in the sense that the basic predicates over polyhedra (emptiness, boundedness, membership, etc) are defined by means of Coq programs.…”

Section: Introductionmentioning

confidence: 99%

A Formalization of Convex Polyhedra Based on the Simplex Method

Allamigeon¹,

Katz²

2017

Interactive Theorem Proving

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We present a formalization of convex polyhedra in the proof assistant Coq. The cornerstone of our work is a complete implementation of the simplex method, together with the proof of its correctness and termination. This allows us to define the basic predicates over polyhedra in an effective way (i.e. as programs), and relate them with the corresponding usual logical counterparts. To this end, we make an extensive use of the Boolean reflection methodology. The benefit of this approach is that we can easily derive the proof of several fundamental results on polyhedra, such as Farkas' Lemma, the duality theorem of linear programming, and Minkowski's Theorem.of the effective nature of our approach is demonstrated by the fact that we easily arrive at the proof of fundamental results on polyhedra, such as several versions of Farkas' Lemma, the duality theorem of linear programming, separation from convex hulls, Minkowski's Theorem, etc.Our effective approach is made possible by implementing the simplex method inside Coq, and proving its correctness and termination. Recall that the simplex method is the first algorithm introduced to solve linear programming [10]. Two difficulties need to be overcome to formalize it. On the one hand, we need to deal with its termination. More precisely, the simplex method iterates over the so-called bases. Its termination depends on the specification of a pivoting rule, whose aim is to determine, at each iteration, the next basis. In this work, we have focused on proving that the lexicographic rule [11] ensures termination. On the other hand, the simplex method is actually composed of two parts. The part previously described, called Phase II, requires an initial basis to start with. Finding such a basis is the purpose of Phase I. It consists in building an extended problem (having a trivial initial basis), and applying to it the Phase II algorithm. Both phases need to be formalized to obtain a fully functional algorithm.We point out that our goal here is not to obtain a practically efficient implementation of the simplex method (e.g., via the code extraction facility of Coq). Rather, we use the simplex method as a tool in our proofs and, in fact, it turns out to be the cornerstone of our approach, given the intuitionistic nature of the logic in Coq. Thus, we adopt the opposite approach of most textbooks on linear programming where, firstly, theoretical results (like the ones mentioned above) are proven, and then the correctness of the simplex method is derived from them.The formalization presented in this paper can be found in a library developed by the authors called Coq-Polyhedra. This library is available through a git repository at https://github.com/nhojem/Coq-Polyhedra. The branch JAR provides the frozen version of the library corresponding to the results presented in this paper. 1 As mentioned above, our formalization is based on the Mathematical Components library [14] (MathComp for short), which is available at https://github.com/math-comp/math-comp. On top of providing a conven...

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Section: Introductionmentioning

confidence: 99%

A Formalization of Convex Polyhedra Based on the Simplex Method

Allamigeon¹,

Katz²

2017

Interactive Theorem Proving

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“…The original Hirsch conjecture, which posited a bound of m − n, was recently disproved for polytopes (i.e. bounded polyhedra) by Santos [15,17], who gave a lower bound of (1 + ε)(m − n) (only slightly violating the conjectured bound).…”

Section: Introductionmentioning

confidence: 99%

On the Shadow Simplex Method for Curved Polyhedra

Dadush

Hähnle

2016

Discrete Comput Geom

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“…Using the fact that in a reduced incident pattern without cycles of length two all vertices must have out-degree at least two, the minimum possible patters were classified in [21]. The proof works without changes for topological prismatoids: It is interesting to note that the prismatoids constructed in [21,24] have the reduced incidence pattern on the left of Figure 2 while the ones we obtain in this paper are related to the pattern on the right. See Section 5 for details.…”

Section: 3mentioning

confidence: 70%

“…Santos' original counterexample applies this result to a 5-prismatoid with 48 vertices and of width six, thus obtaining a non-Hirsch 23-polytope with 46 facets. This was improved in [21] to a 5-prismatoid of the same width but with only 25 vertices, which provides a non-Hirsch polytope in dimension 20.…”

Section: 3mentioning

confidence: 99%

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Topological Prismatoids and Small Simplicial Spheres of Large Diameter

Criado

Santos

2019

Experimental Mathematics

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We introduce topological prismatoids, a combinatorial abstraction of the (geometric) prismatoids recently introduced by the second author to construct counter-examples to the Hirsch conjecture. We show that the "strong d-step Theorem" that allows to construct such large-diameter polytopes from "non-d-step" prismatoids still works at this combinatorial level. Then, using metaheuristic methods on the flip graph, we construct four combinatorially different non-d-step 4-dimensional topological prismatoids with 14 vertices. This implies the existence of 8-dimensional spheres with 18 vertices whose combinatorial diameter exceeds the Hirsch bound. These examples are smaller that the previously known examples by Mani and Walkup in 1980 (24 vertices, dimension 11).Our non-Hirsch spheres are shellable but we do not know whether they are realizable as polytopes.2000 Mathematics Subject Classification. 52B05, 52B12, 90C60, 90C05.

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The width of five-dimensional prismatoids

Cited by 27 publications

References 11 publications

A Formalization of Convex Polyhedra Based on the Simplex Method

A Formalization of Convex Polyhedra Based on the Simplex Method

On the Shadow Simplex Method for Curved Polyhedra

Topological Prismatoids and Small Simplicial Spheres of Large Diameter

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