arXiv subjects

Francisco Criado

Publications and source records attributed to Francisco Criado.

11 recordsLinked to original sources

arXiv2026

Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture

Henze and Malikiosis (2017) have shown that the Lonely Runner Conjecture (LRC) can be restated as a convex-geometric question on the so-called LR zonotopes, lattice zonotopes with one more generator than their dimension. This relation naturally suggests a more generel statement, the "shifted" LRC, the zonotopal version of which concerns a classical parameter, the covering radius. Theorems A and B in Malikiosis-Schymura-Santos (2025) use the zonotopal restatements of both the original and the shifted LRC to prove a linearly-exponential bound on the size of the (integer) speeds for which the conjectures need to be checked in order to establish them for each fixed number of runners; in the shifted version their statement and proof rely on a certain assumption on two-dimensional rational vector configurations, the so-called "Lonely Vector Property". In this paper we do two things: We push the analogies between the two versions of LRC and their zonotopal counterparts, in particular highlighting that the proofs of Theorems A and B in Malikiosis-Schymura-Santos are more transparent, and the statments more general, if regarded in terms of two quite general classes of lattice zonotopes: the coloopless zonotopes that we introduce here and the cosimple ones, already defined by them. These classes contain all primitive zonotopes of widths at least two and at least three, respectively. We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at $n=5$) and to the Lonely Vector Property (starting at $n=12$).

math.CO↗

arXiv2025

Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $μ$-Polyak-Łojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.

math.OC↗

arXiv2025

Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture

We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes (Henze and Malikiosis, 2017), and on an upper bound for the (integer) velocities to be checked (Malikiosis, Santos and Schymura, 2024+). As a tool for the proof, we devise an algorithm for bounding the covering radius of rational lattice polytopes, based on constructing dyadic fundamental domains.

math.CO↗

arXiv2025

On the convex hull of integer points above the hyperbola

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $Ω(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational slopes except that instead of $n$ we have $n/Δ$ in the lower bound and by $\max\left\{Δ, n/Δ\right\}$ in the upper bound, where $Δ\in \mathbb{Z}_{>0}$ is the discriminant. We also give an algorithm that enumerates the vertices of these convex hulls in logarithmic time per vertex. One motivation for such an algorithm is the deterministic factorization of integers.

math.CO↗

arXiv2025

The algorithmic Fried Potato Problem in two dimensions

Conway's Fried Potato Problem seeks to determine the best way to cut a convex body in $n$ parts by $n-1$ hyperplane cuts (with the restriction that the $i$-th cut only divides in two one of the parts obtained so far), in a way as to minimize the maxuimum of the inradii of the parts. It was shown by Bezdek and Bezdek that the solution is always attained by $n-1$ parallel cuts. But the algorithmic problem of finding the best direction for these parallel cuts remains. In this note we show that for a convex $m$-gon $P$, this direction (and hence the cuts themselves) can be found in time $O(m \log^4 m)$, which improves on a quadratic algorithm proposed by Cañete-Fernández-Márquez (DMD 2022). Our algorithm first preprocesses what we call the dome (closely related to the medial axis) of $P$ using a variant of the Dobkin-Kirkpatrick hierarchy, so that linear programs in the dome and in slices of it can be solved in polylogarithmic time.

math.MG↗

arXiv2024

Random Shadows of Fixed Polytopes

Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove multiple upper bounds on the expected number of vertices of a random shadow of a fixed polytope. Our bounds are in terms of various parameters in the literature including geometric diameter and edge lengths, minimal and maximal slack, maximal coordinates for lattice polytopes, and maximum absolute values of subdeterminants. For the case of geometric diameter and edge lengths, we prove lower bounds and argue that our upper and lower bounds are both tight for zonotopes.

math.CO↗

arXiv2022

Fast Algorithms for Packing Proportional Fairness and its Dual

The proportional fair resource allocation problem is a major problem studied in flow control of networks, operations research, and economic theory, where it has found numerous applications. This problem, defined as the constrained maximization of $\sum_i \log x_i$, is known as the packing proportional fairness problem when the feasible set is defined by positive linear constraints and $x \in \mathbb{R}^{n}_{\geq 0}$. In this work, we present a distributed accelerated first-order method for this problem which improves upon previous approaches. We also design an algorithm for the optimization of its dual problem. Both algorithms are width-independent. Finally, we show the latter problem has applications to the volume reduction of bounding simplices in an old linear programming algorithm of (YL82), and we obtain some improvements as a result.

math.OC↗

arXiv2021

Tropical bisectors and Voronoi diagrams

In this paper we initiate the study of tropical Voronoi diagrams. We start out with investigating bisectors of finitely many points with respect to arbitrary polyhedral norms. For this more general scenario we show that bisectors of three points are homeomorphic to a non-empty open subset of Euclidean space, provided that certain degenerate cases are excluded. Specializing our results to tropical bisectors then yields structural results and algorithms for tropical Voronoi diagrams.

math.CO↗

arXiv2019

Topological Prismatoids and Small Simplicial Spheres of Large Diameter

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.

math.CO↗

arXiv2019

Randomized construction of complexes with large diameter

We consider the question of the largest possible combinatorial diameter among $(d-1)$-dimensional simplicial complexes on $n$ vertices, denoted $H_s(n, d)$. Using a probabilistic construction we give a new lower bound on $H_s(n, d)$ that is within an $O(d^2)$ factor of the upper bound. This improves on the previously best-known lower bound which was within a factor of $e^{Θ(d)}$ of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

math.CO↗

arXiv2016

The maximum diameter of pure simplicial complexes and pseudo-manifolds

We construct $d$-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with $n$ vertices whose combinatorial diameter grows as $c_d n^{d-1}$ for a constant $c_d$ depending only on $d$, which is the maximum possible growth. Moreover, the constant $c_d$ is optimal modulo a singly exponential factor in $d$. The pure simplicial complexes improve on a construction of the second author that achieved $c_d n^{2d/3}$. For pseudo-manifolds without boundary, as far as we know, no construction with diameter greater than $n^2$ was previously known.

math.CO↗