arXiv subjects

Antoine Deza

Publications and source records attributed to Antoine Deza.

At least 37 records · Page 2Linked to original sources

arXiv2018

Optimization over Degree Sequences

We introduce and study the problem of optimizing arbitrary functions over degree sequences of hypergraphs and multihypergraphs. We show that over multihypergraphs the problem can be solved in polynomial time. For hypergraphs, we show that deciding if a given sequence is the degree sequence of a 3-hypergraph is NP-complete, thereby solving a 30 year long open problem. This implies that optimization over hypergraphs is hard already for simple concave functions. In contrast, we show that for graphs, if the functions at vertices are the same, then the problem is polynomial time solvable. We also provide positive results for convex optimization over multihypergraphs and graphs and exploit connections to degree sequence polytopes and threshold graphs. We then elaborate on connections to the emerging theory of shifted combinatorial optimization.

math.OC↗

arXiv2018

Diameter, decomposability, and Minkowski sums of polytopes

We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes P and Q, we show that P can be written as a Minkowski sum with a summand homothetic to Q if and only if P has the same number of vertices as its Minkowski sum with Q.

math.MG↗

arXiv2017

Managing losses in exotic horse race wagering

We consider a specialized form of risk management for betting opportunities with low payout frequency, presented in particular for exotic horse race wagering. An optimization problem is developed which limits losing streaks with high probability to the given time horizon of a gambler, which is formulated as a globally solvable mixed integer non-linear program. A case study is conducted using one season of historical horse racing data.

math.OC↗

arXiv2017

Computational determination of the largest lattice polytope diameter

A lattice (d, k)-polytope is the convex hull of a set of points in dimension d whose coordinates are integers between 0 and k. Let δ(d, k) be the largest diameter over all lattice (d, k)-polytopes. We develop a computational framework to determine δ(d, k) for small instances. We show that δ(3, 4) = 7 and δ(3, 5) = 9; that is, we verify for (d, k) = (3, 4) and (3, 5) the conjecture whereby δ(d, k) is at most (k + 1)d/2 and is achieved, up to translation, by a Minkowski sum of lattice vectors.

cs.CG↗

arXiv2017

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.

math.OC↗

arXiv2016

Improved bounds on the diameter of lattice polytopes

We show that the largest possible diameter $δ(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that $δ(4,3)=8$. This substantiates the conjecture whereby $δ(d,k)$ is at most $\lfloor(k+1)d/2\rfloor$ and is achieved by a Minkowski sum of lattice vectors.

math.MG↗

arXiv2015

Imperfect demand estimation for new product production planning

We are interested in the effect of consumer demand estimation error for new products in the context of production planning. An inventory model is proposed, whereby demand is influenced by price and advertising. The effect of parameter misspecification of the demand model is empirically examined in relation to profit and service level feasibility. Faced with an uncertain consumer reaction to price and advertising, we find that it is safer to overestimate rather than underestimate the effect of price on demand. Moreover, under a service level constraint it is safer to overestimate the effect of advertising, whereas for strict profit maximization, underestimating the effect of advertising is the conservative approach.

math.OC↗

arXiv2014

A primal-simplex based Tardos' algorithm

In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin's primal-based modification and Mizuno's use of the simplex method, we introduce a modification of Tardos' algorithm considering only the primal problem and using simplex method to solve the auxiliary problems. The proposed algorithm is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate.

math.OC↗

arXiv2014

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 2n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2n - Theta(log n). We show that a string of length n contains at most 5n/3 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 2n/3 double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.

math.CO↗

arXiv2014

Chance Constrained Optimization for Targeted Internet Advertising

We introduce a chance constrained optimization model for the fulfillment of guaranteed display Internet advertising campaigns. The proposed formulation for the allocation of display inventory takes into account the uncertainty of the supply of Internet viewers. We discuss and present theoretical and computational features of the model via Monte Carlo sampling and convex approximations. Theoretical upper and lower bounds are presented along with a numerical substantiation.

cs.CE↗

arXiv2014

A further generalization of the colourful Carathéodory theorem

Given $d+1$ sets, or colours, $S_1, S_2,...,S_{d+1}$ of points in $\mathbb{R}^d$, a {\em colourful} set is a set $S\subseteq\bigcup_i S_i$ such that $|S\cap S_i|\leq 1$ for $i=1,...,d+1$. The convex hull of a colourful set $S$ is called a {\em colourful simplex}. Bárány's colourful Carathéodory theorem asserts that if the origin 0 is contained in the convex hull of $S_i$ for $i=1,...,d+1$, then there exists a colourful simplex containing 0. The sufficient condition for the existence of a colourful simplex containing 0 was generalized to 0 being contained in the convex hull of $S_i\cup S_j$ for $1\leq i< j \leq d+1$ by Arocha et al. and by Holmsen et al. We further generalize the sufficient condition and obtain new colourful Carathéodory theorems. We also give an algorithm to find a colourful simplex containing 0 under the generalized condition. In the plane an alternative, and more general, proof using graphs is given. In addition, we observe that any condition implying the existence of a colourful simplex containing 0 actually implies the existence of $\min_i|S_i|$ such simplices.

cs.CG↗

arXiv2013

A combinatorial approach to colourful simplicial depth

The colourful simplicial depth conjecture states that any point in the convex hull of each of d+1 sets, or colours, of d+1 points in general position in R^d is contained in at least d^2+1 simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which can not arise from a colourful point configuration. The number of octahedral systems is also given.

math.CO↗

arXiv2012

Computational Lower Bounds for Colourful Simplicial Depth

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d^2+1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d=3, however the best known lower bound for d at least 4 is ((d+1)^2)/2. A promising method to improve this lower bound is to look at combinatorial octahedral systems generated by such configurations. The difficulty to employing this approach is handling the many symmetric configurations that arise. We propose a table of invariants which exclude many of partial configurations, and use this to improve the lower bound in dimension 4.

math.CO↗

arXiv2010

More Colourful Simplices

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

math.CO↗

arXiv2009

More bounds on the diameters of convex polytopes

Finding a good bound on the maximal edge diameter $Δ(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known, the behaviour of the function $Δ(d,n)$ is largely unknown. The Hirsch conjecture, formulated in 1957 and reported in \cite{GD}, states that $Δ(d,n)$ is linear in $n$ and $d$: $Δ(d,n) \leq n-d$. The conjecture is known to hold in small dimensions, i.e., for $d \leq 3$ \cite{VK}, along with other specific pairs of $d$ and $n$ (Table \ref{before}). However, the asymptotic behaviour of $Δ(d,n)$ is not well understood: the best upper bound -- due to Kalai and Kleitman -- is quasi-polynomial \cite{GKDK}. In this article we will show that $Δ(4,12)=7$ and present strong evidence for $Δ(5,12)=Δ(6,13)=7$. The first of these new values is of particular interest since it indicates that the Hirsch bound is not sharp in dimension 4.

math.CO↗

arXiv2007

Hyperplane Arrangements with Large Average Diameter

The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed dimension. We give the exact value of the largest possible average diameter for all simple arrangements in dimension 2, for arrangements having at most the dimension plus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds which are both asymptotically equal to the dimension.

math.MG↗

arXiv2007

The complexity of the envelope of line and plane arrangements

A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is hypothesized to be at least $d{n-2 \choose d-1}$. In this note we show that, for simple arrangements of 4 lines or more, the minimum number of external facets is equal to $2(n-1)$, and for simple arrangements of 5 planes or more, the minimum number of external facets is between $\frac{n(n-2)+6}{3}$ and $(n-4)(2n-3)+5$.

math.MG↗

arXiv2006

Colourful Simplicial Depth

Inspired by Barany's colourful Caratheodory theorem, we introduce a colourful generalization of Liu's 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 d^2+1 and that the maximum is d^(d+1)+1. We exhibit configurations attaining each of these depths and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.

math.CO↗