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Cristiano Bocci

Publications and source records attributed to Cristiano Bocci.

At least 19 recordsLinked to original sources
arXiv2024

A New Temporal Interpretation of Cluster Editing

The NP-complete graph problem Cluster Editing seeks to transform a static graph into a disjoint union of cliques by making the fewest possible edits to the edges. We introduce a natural interpretation of this problem in temporal graphs, whose edge sets change over time. This problem is NP-complete even when restricted to temporal graphs whose underlying graph is a path, but we obtain two polynomial-time algorithms for restricted cases. In the static setting, it is well-known that a graph is a disjoint union of cliques if and only if it contains no induced copy of $P_3$; we demonstrate that no general characterisation involving sets of at most four vertices can exist in the temporal setting, but obtain a complete characterisation involving forbidden configurations on at most five vertices. This characterisation gives rise to an FPT algorithm parameterised simultaneously by the permitted number of modifications and the lifetime of the temporal graph.

cs.DM↗
arXiv2022

Hadamard products of hypersurfaces

In this paper we, first, characterize hypersurfaces for which their Hadamard product is still a hypersurface. Then we pass to study hypersurfaces and, more generally, varieties which are idempotent under Hadamard powers.

math.AG↗
arXiv2022

Gorenstein points in $\mathbb{P}^3$ via Hadamard product of projective varieties

We show how to construct a stick figure of lines in $\mathbb{P}^3$ using the Hadamard product of projective varieties. Then, applying the results of Migliore and Nagel, we use such stick figure to build a Gorenstein set of points with given $h-$vector ${\mathbf h}$. Since the Hadamard product is a coordinate-wise product, we show, at the end, how the coordinates of the points, in the Gorenstein set, can be directly determined.

math.AG↗
arXiv2020

Realization of distance matrices by graphs of genus 1

Given a distance matrix $D$, we study the behavior of its compaction vector and reduction matrix with respect to the problem of the realization of $D$ by a weighted graph. To this end, we first give a general result on realization by $n-$cycles and successively we mainly focus on graphs of genus 1, presenting an algorithm which determines when a distance matrix is realizable by such a kind of graph, and then, shows how to construct it.

math.CO↗
arXiv2020

Catalecticant intersections and confinement of decompositions of forms

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (\cite{London90}) that $3$ general plane cubics have $2$ simultaneous Waring decompositions of rank $6$. Then we compute, with the software Bertini, that $4$ general plane quartics have $18$ different decompositions of rank $10$ (a result which was not known before).

math.AG↗
arXiv2018

Algebra and geometry of tensors for modeling rater agreement data

We study three different quasi-symmetry models and three different mixture models of $n\times n\times n$ tensors for modeling rater agreement data. For these models we give a geometric description of the associated varieties and we study their invariants distinguishing between the case $n=2$ and the case $n>2$. Finally, for the two models for pairwise agreement we state some results about the pairwise Cohen's $κ$ coefficients.

math.ST↗
arXiv2017

Exact tests to compare contingency tables under quasi-independence and quasi-symmetry

In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the use of these models in different fields of applications.

math.ST↗
arXiv2016

Real identifiability vs complex identifiability

Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it has a unique decomposition in terms of rank $1$ tensors. There are cases in which the identifiability fails over the complex field, for general tensors of rank $r$. This behavior is quite peculiar when the rank $r$ is submaximal. Often, the failure is due to the existence of an elliptic normal curve through general points of the corresponding Segre, Veronese or Grassmann variety. We prove the existence of nonempty euclidean open subsets of some variety of tensors of rank $r$, whose elements have several decompositions over $\mathbb C$, but only one of them is formed by real summands. Thus, in the open sets, tensors are not identifiable over $\mathbb C$, but are identifiable over $\mathbb R$. We also provide examples of non trivial euclidean open subsets in a whole space of symmetric tensors (of degree $7$ and $8$ in three variables) and of almost unbalanced tensors Segre Product ($\mathbb P^2\times \mathbb P^4\times \mathbb P^9$) whose elements have typical real rank equal to the complex rank, and are identifiable over $\mathbb R$, but not over $\mathbb C$. On the contrary, we provide examples of tensors of given real rank, for which real identifiability cannot hold in non-trivial open subsets.

math.AG↗
arXiv2016

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal $I \subseteq R =k[x_1,\ldots,x_n]$, we show that $\widehatα(I)$, the Waldschmidt constant of $I$, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of $I$. By applying results from fractional graph theory, we can then express $\widehatα(I)$ in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of $I$. Moreover, expressing $\widehatα(I)$ as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on $\widehatα(I)$, thus verifying a conjecture of Cooper-Embree-Hà-Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of $\mathbb{P}^n$ with few components compared to $n$, and we find the Waldschmidt constant for the Stanley-Reisner ideal of a uniform matroid.

math.AC↗
arXiv2015

Hadamard Products of Linear Spaces

We describe properties of Hadamard products of algebraic varieties. We show any Hadamard power of a line is a linear space, and we construct star configurations from products of collinear points. Tropical geometry is used to find the degree of Hadamard products of other linear spaces.

math.AG↗
arXiv2013

Containment results for ideals of various configurations of points in P^N

Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in ${\bf P}^N$. In this paper we propose another conjecture along the same lines (Conjecture 3.9), and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points.

math.AG↗
arXiv2013

Refined methods for the identifiability of tensors

We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.9997 (2^n)/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.998 (3^n)/(2n+1) (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.

math.AG↗
arXiv2011

Max-plus objects to study the complexity of graphs

Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of $H_G$ in terms of the adjacency matrix of $G$ and we give a central limit theorem for $H_G$. Finally, we show that the mp-chart is easily tractable also for the complement graph.

math.PR↗
arXiv2011

On the identifiability of binary Segre products

We prove that a product of $m>5$ copies of $\PP^1$, embedded in the projective space $\PP^r$ by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant $k$-space), for all $k$ such that $k+1\leq 2^{m-1}/m$.

math.AG↗
arXiv2011

Recent developments and open problems in linear series

In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.

math.AG↗
arXiv2009

Geometry of diagonal-effect models for contingency tables

In this work we study several types of diagonal-effect models for two-way contingency tables in the framework of Algebraic Statistics. We use both toric models and mixture models to encode the different behavior of the diagonal cells. We compute the invariants of these models and we explore their geometrical structure.

math.ST↗