arXiv subjects
Publications and source records attributed to Hebert Pérez-Rosés.
The change-making problem was recently extended to sets of positive integers not containing the element $1$, and from there to numerical semigroups. A greedy numerical semigroup is defined as a numerical semigroup where the greedy representation of an element is optimal with respect to the number of summands. In this paper we identify some new families of greedy numerical semigroups.
We make an asymptotic analysis via singularity analysis of generating functions of a number sequence that involves the Fibonacci numbers and generalizes the binomial coefficients.
Motivated by the change-making problem, we extend the notion of greediness to sets of positive integers not containing the element $1$, and from there to numerical semigroups. We provide an algorithm to determine if a given set (not necessarily containing the number $1$) is greedy. We also give specific conditions for sets of cardinality three, and we prove that numerical semigroups generated by three consecutive integers are greedy.
The change-making problem consists of representing a certain amount of money with the least possible number of coins, from a given, pre-established set of denominations. The greedy algorithm works by choosing the coins of largest possible denomination first. This greedy strategy does not always produce the least number of coins, except when the set of denominations obeys certain properties. We call a set of denominations with these properties a greedy set. If the set of denominations is an infinite sequence, we call it totally greedy if every prefix subset is greedy. In this paper we investigate some totally greedy sequences arising from second-order linear recurrences with constant coefficients, as well as their subsequences, and we prove sufficient conditions under which these sequences are totally greedy.
We introduce the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups with the given genus. This construction provides an alternative approach to the conjecture on the increasingness of the number of numerical semigroups for each given genus. We elaborate on the number of nodes at each tree depth in the forest and present a few new conjectures that can be developed in the future. We prove some properties of the quasi-ordinarization transform, its relations with the ordinarization transform, and we also present an alternative approach to the conjecture that the number of numerical semigroups of each given genus is increasing.
We introduce the inverse Voronoi diagram problem in graphs: given a graph $G$ with positive edge-lengths and a collection $\mathbb{U}$ of subsets of vertices of $V(G)$, decide whether $\mathbb{U}$ is a Voronoi diagram in $G$ with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph. For trees we show that the problem can be solved in $O(N+n \log^2 n)$ time, where $n$ is the number of vertices in the tree and $N=n+\sum_{U\in \mathbb{U}}|U|$ is the size of the description of the input. We also provide a lower bound of $Ω(n \log n)$ time for trees with $n$ vertices.
A vertex $v$ of a graph $G=(V,E)$ is said to be undefended with respect to a function $f: V \longrightarrow \{0,1,2\}$ if $f(v)=0$ and $f(u)=0$ for every vertex $u$ adjacent to $v$. We call the function $f$ a weak Roman dominating function if for every $v$ such that $f(v)=0$ there exists a vertex $u$ adjacent to $v$ such that $f(u)\in \{1,2\}$ and the function $f': V \longrightarrow \{0,1,2\}$ defined by $f'(v)=1$, $f'(u)=f(u)-1$ and $f'(z)=f(z)$ for every $z\in V \setminus\{u,v\}$, has no undefended vertices. The weight of $f$ is $w(f)=\sum_{v\in V(G) }f(v)$. The weak Roman domination number of a graph $G$, denoted by $γ_r(G)$, is the minimum weight among all weak Roman dominating functions on $G$. Henning and Hedetniemi [Discrete Math. 266 (2003) 239-251] showed that the problem of computing $γ_r(G)$ is NP-Hard, even when restricted to bipartite or chordal graphs. This suggests finding $γ_r(G)$ for special classes of graphs or obtaining good bounds on this invariant. In this article, we obtain closed formulae and tight bounds for the weak Roman domination number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product.
The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees.
Some social networks, such as LinkedIn and ResearchGate, allow user endorsements for specific skills. In this way, for each skill we get a directed graph where the nodes correspond to users' profiles and the arcs represent endorsement relations. From the number and quality of the endorsements received, an authority score can be assigned to each profile. In this paper we propose an authority score computation method that takes into account the relations existing among different skills. Our method is based on enriching the information contained in the digraph of endorsements corresponding to a specific skill, and then applying a ranking method admitting weighted digraphs, such as PageRank. We describe the method, and test it on a synthetic network of 1493 nodes, fitted with endorsements.
In many simulation studies involving networks there is the need to rely on a sample network to perform the simulation experiments. In many cases, real network data is not available due to privacy concerns. In that case we can recourse to synthetic data sets with similar properties to the real data. In this paper we discuss the problem of generating synthetic data sets for a certain kind of online social network, for simulation purposes. Some popular online social networks, such as LinkedIn and ResearchGate, allow user endorsements for specific skills. For each particular skill, the endorsements give rise to a directed subgraph of the corresponding network, where the nodes correspond to network members or users, and the arcs represent endorsement relations. Modelling these endorsement digraphs can be done by formulating an optimization problem, which is amenable to different heuristics. Our construction method consists of two stages: The first one simulates the growth of the network, and the second one solves the aforementioned optimization problem to construct the endorsements.
Interfaces based on projection screens have become increasingly more popular in recent years, mainly due to the large screen size and resolution that they provide, as well as their stereo-vision capabilities. This work shows a local method for real-time detection of non-stationary photometric perturbations in projected images by means of computer vision techniques. The method is based on the computation of differences between the images in the projector's frame buffer and the corresponding images on the projection screen observed by the camera. It is robust under spatial variations in the intensity of light emitted by the projector on the projection surface and also robust under stationary photometric perturbations caused by external factors. Moreover, we describe the experiments carried out to show the reliability of the method.
Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ "fits" \ $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.
Graph labellings have been a very fruitful area of research in the last four decades. However, despite the staggering number of papers published in the field (over 1000), few general results are available, and most papers deal with particular classes of graphs and methods. Here we approach the problem from the computational viewpoint, and in a quite general way. We present the existence problem of a particular labelling as a combinatorial optimization problem, then we discuss the possible strategies to solve it, and finally we present a heuristic for finding different classes of labellings, like vertex-, edge-, or face-magic, and $(a, d)$-antimagic $(v, e, f)$-labellings. The algorithm has been implemented in C++ and MATLAB, and with its aid we have been able to derive new results for some classes of graphs, in particular, vertex-antimagic edge labellings for small graphs of the type $P_2^r \times P_3^s$, for which no general construction is known so far.