Antoine Deza

The Ridge Graph of the Metric Polytope and Some Relatives

The metric polytope is a \(\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array}} \right) \) -dimensional convex polytope defined by its 4 \(\left( {\begin{array}{*{20}c} n \\ 3 \\ \end{array}} \right) \) facets. The vertices of the metric polytope are known only up to n = 6, for n = 7 they number more than 60 000. The study of the metric polytope and its relatives (the metric cone, the cut polytope and the cut cone) is mainly motivated by their application to the maximum cut and multicommodity flow feasibility problems. We characterize the ridge graph of the metric polytope, i.e. the edge graph of its dual, and, as corollary, obtain that the diameter of the dual metric polytope is 2. For n ≥ 5, the edge graph of the metric polytope restricted to its integral vertices called cuts, and to some \(\left\{ {\frac{1}{3},\,\frac{2}{3}} \right\} \) -valued vertices called anticuts, is, besides the clique on the cuts, the bipartite double of the complement of the folded n-cube. We also give similar results for the metric cone, the cut polytope and the cut cone.

Polytopes and arrangements: Diameter and curvature

We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub. We prove a continuous analogue of the result of Holt and Klee, namely, we construct a family of polytopes which attain the conjectured order of the largest total curvature.

Colourful Simplicial Depth

AbstractInspired by Barany’s Colourful Caratheodory Theorem, we introduce a colourful generalization of Lius simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.

On the Skeleton of the Metric Polytope

We consider convex polyhedra with applications to well-known combinatorial optimization problems: the metric polytope mn and its relatives. For n ? 6 the description of the metric polytope is easy as mn has at most 544 vertices partitioned into 3 orbits; m7 - the largest previously known instance - has 275 840 vertices but only 13 orbits. Using its large symmetry group, we enumerate orbitwise 1 550 825 600 vertices of the 28-dimensional metric polytope m8. The description consists of 533 orbits and is conjectured to be complete. The orbitwise incidence and adjacency relations are also given. The skeleton of m8 could be large enough to reveal some general features of the metric polytope on n nodes. While the extreme connectivity of the cuts appears to be one of the main features of the skeleton of mn, we conjecture that the cut vertices do not form a cut-set. The combinatorial and computational applications of this conjecture are studied. In particular, a heuristic skipping the highest degeneracy is presented.

Fullerenes and coordination polyhedra versus half-cube embeddings

A fullerene Fn is a 3-regular (or cubic) polyhedral carbon molecule for which the n vertices - the carbons atoms - are arranged in 12 pentagons and (n/2 - 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes Fn for n < 60 and of all preferable fullerenes Cn for n < 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onion-like metallic clusters and geodesic domes. Quasi-embeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells.

How many double squares can a string contain

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson?showed in 1998 that a string of length n contains at most 2 n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2 n - ? ( log n ) . We show that a string of length n contains at most ? 11 n / 6 ? distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most ? 5 n / 6 ? particular double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpsons result.

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes

We consider a family of linear optimization problems over the n-dimensional Klee—Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2 n –2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2 n ) iterations to solve this problem, whereas in practice typically only a few iterations (e.g., 50) suffices to obtain a high-quality solution. Thus, the construction potentially exhibits the worst-case iteration-complexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n).

How good are interior point methods? Klee–Minty cubes tighten iteration-complexity bounds

By refining a variant of the Klee–Minty example that forces the central path to visit all the vertices of the Klee–Minty n-cube, we exhibit a nearly worst-case example for path-following interior point methods. Namely, while the theoretical iteration-complexity upper bound is

A Continuous d -Step Conjecture for Polytopes

O(2^{n}n^{\frac{5}{2}})