arXiv subjects

Josep Díaz

Publications and source records attributed to Josep Díaz.

11 recordsLinked to original sources

arXiv2025

The Metric Dimension of Sparse Random Graphs

In 2013, Bollobás, Mitsche, and Pralat at gave upper and lower bounds for the likely metric dimension of random Erdős-Rényi graphs $G(n,p)$ for a large range of expected degrees $d=pn$. However, their results only apply when $d \ge \log^5 n$, leaving open sparser random graphs with $d < \log^5 n$. Here we provide upper and lower bounds on the likely metric dimension of $G(n,p)$ from just above the connectivity transition, i.e., where $d=pn=c \log n$ for some $c > 1$, up to $d=\log^5 n$. Our lower bound technique is based on an entropic argument which is more general than the use of Suen's inequality by Bollobás, Mitsche, and Pralat, whereas our upper bound is similar to theirs.

math.CO↗

arXiv2025

Rainbow connectivity of multilayered random geometric graphs

An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from $h\ge 2$ independent random geometric graphs $G(n,r)$ on the unit square. We define an edge-coloring by coloring the edges according to the copy of $G(n,r)$ they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that $r(n)=\left(\frac{\log n}{n}\right)^{\frac{h-1}{2h}}$ is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erdős-R\' enyi random model.

math.CO↗

arXiv2024

Dynamic random graphs with vertex removal

We introduce and analyse a Dynamic Random Graph with Vertex Removal (DRGVR) defined as follows. At every step, with probability $p > 1/2$ a new vertex is introduced, and with probability $1-p$ a vertex, chosen uniformly at random among the present ones (if any), is removed from the graph together with all edges adjacent to it. In the former case, the new vertex connects by an edge to every other vertex with probability inversely proportional to the number of vertices already present. We prove that the DRGVR converges to a local limit and determine this limit. Moreover, we analyse its component structure and distinguish a subcritical and a supercritical regime with respect to the existence of a giant component. As a byproduct of this analysis, we obtain upper and lower bounds for the critical parameter. Furthermore, we provide precise expression of the maximum degree (as well as in- and out-degree for a natural orientation of the DRGVR). Several concentration and stability results complete the study.

math.PR↗

arXiv2022

On Vertex Bisection Width of Random $d$-Regular Graphs

Vertex bisection is a graph partitioning problem in which the aim is to find a partition into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. We are interested in giving upper bounds on the vertex bisection width of random $d$-regular graphs for constant values of $d$. Our approach is based on analyzing a greedy algorithm by using the Differential Equations Method. In this way, we obtain the first known upper bounds for the vertex bisection width in random regular graphs. The results are compared with experimental ones and with lower bounds obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers of Random Regular Graphs, SIAM J. on Disc. Math. 28(1), 553-575, 2014).

cs.DS↗

arXiv2022

Reconstruction of Random Geometric Graphs: Breaking the Omega(r) distortion barrier

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where $n$ points are scattered uniformly in a square of area $n$, and two points have an edge between them if and only if their Euclidean distance is less than $r$. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if $r=n^α$ for any $α> 0$, with high probability reconstructs the vertex positions with a maximum error of $O(n^β)$ where $β=1/2-(4/3)α$, until $α\ge 3/8$ where $β=0$ and the error becomes $O(\sqrt{\log n})$. This improves over earlier results, which were unable to reconstruct with error less than $r$. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in $\R^3$ and to hypercubes in any constant fixed dimension. Additionally we examine the extent to which reconstruction is still possible when the original adjacency lists have had a subset of the edges independently deleted at random.

cs.CG↗

arXiv2021

The Multicolored Graph Realization Problem

We introduce the Multicolored Graph Realization problem (MGRP). The input to the problem is a colored graph $(G,φ)$, i.e., a graph together with a coloring on its vertices. We can associate to each colored graph a cluster graph ($G_φ)$ in which, after collapsing to a node all vertices with the same color, we remove multiple edges and self-loops. A set of vertices $S$ is multicolored when $S$ has exactly one vertex from each color class. The problem is to decide whether there is a multicolored set $S$ such that, after identifying each vertex in $S$ with its color class, $G[S]$ coincides with $G_φ$. The MGR problem is related to the class of generalized network problems, most of which are NP-hard. For example the generalized MST problem. MGRP is a generalization of the Multicolored Clique Problem, which is known to be W[1]-hard when parameterized by the number of colors. Thus MGRP remains W[1]-hard, when parameterized by the size of the cluster graph and when parameterized by any graph parameter on $G_φ$, among those for treewidth. We look to instances of the problem in which both the number of color classes and the treewidth of $G_φ$ are unbounded. We show that MGRP is NP-complete when $G_φ$ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our hardness results follows from suitable reductions from the 1-in-3 monotone SAT problem. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds also graphs with bounded degree. We complement those results by showing combined parameterizations under which the MGR problem became tractable.

cs.CC↗

arXiv2020

On List k-Coloring Convex Bipartite Graphs

List k-Coloring (Li k-Col) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..,k}. The problem is known to be NP-hard even for k=3 within the class of 3-regular planar bipartite graphs and for k=4 within the class of chordal bipartite graphs. In 2015, Huang, Johnson and Paulusma asked for the complexity of Li 3-Col in the class of chordal bipartite graphs. In this paper we give a partial answer to this question by showing that Li k-Col is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the Li k-Col in the calss of convex bipartite graphs. Finally we show how our algorithm can be modified to solve the more general Li H-Col problem on convex bipartite graphs.

cs.CC↗

arXiv2019

Algorithmically Efficient Syntactic Characterization of Possibility Domains

In the field of Judgment Aggrgation, a domain, that is a subset of a Cartesian power of $\{0,1\}$, is considered to reflect abstract rationality restrictions on vectors of two-valued judgments on a number of issues. We are interested in the ways we can aggregate the positions of a set of individuals, whose positions over each issue form vectors of the domain, by means of unanimous (idempotent) functions, whose output is again an element of the domain. Such functions are called non-dictatorial, when their output is not simply the positions of a single individual. Here, we consider domains admitting various kinds of non-dictatorial aggregators, which reflect various properties of majority aggregation: (locally) non-dictatorial, generalized dictatorships, anonymous, monotone, StrongDem and systematic. We show that interesting and, in some sense, democratic voting schemes are always provided by domains that can be described by propositional formulas of specific syntactic types we define. Furthermore, we show that we can efficiently recognize such formulas and that, given a domain, we can both efficiently check if it is described by such a formula and, in case it is, construct it. Our results fall in the realm of classical results concerning the syntactic characterization of domains with specific closure properties, like domains closed under logical AND which are the models of Horn formulas. The techniques we use to obtain our results draw from judgment aggregation as well as propositional logic and universal algebra.

cs.CC↗

arXiv2014

Optimizing the Social Cost of Congestion Games by Imposing Variable Delays

We describe a new coordination mechanism for non-atomic congestion games that leads to a (selfish) social cost which is arbitrarily close to the non-selfish optimal. This mechanism does not incur any additional extra cost, like tolls, which are usually differentiated from the social cost as expressed in terms of delays only.

cs.GT↗

arXiv2014

On the relation between graph distance and Euclidean distance in random geometric graphs

Given any two vertices u, v of a random geometric graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention in the literature. In this paper, we improve these upper bounds for values of r=omega(sqrt(log n)) (i.e. for r above the connectivity threshold). Our result also improves the best-known estimates on the diameter of random geometric graphs. We also provide a lower bound on d_G(u,v) in terms of d_E(u,v).

cs.DM↗

arXiv2014

Approximating Fixation Probabilities in the Generalized Moran Process

We consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312--316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned 'fitness' value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness $r>0$ placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when $r\geq 1$) and of extinction (for all $r>0$).

cs.CC↗