arXiv subjects

Pablo Romero

Publications and source records attributed to Pablo Romero.

17 recordsLinked to original sources

arXiv2025

Construction of infinitely many trace-minimal graphs with maximum number of spanning trees

A longstanding problem in spectral graph theory asks for graphs with maximum number of spanning trees among all connected simple graphs with a prescribed number of vertices and edges. Such graphs are called t-optimal graphs. Petingi and Rodr\'iguez [Discrete Math. 244 (2002), 351--373] achieved in finding infinitely many t-optimal graphs. Basically, they reduced the problem of finding t-optimal graphs to the determination of almost-regular graphs with minimum number of induced 3-paths. In this work we revisit the construction of t-optimal graphs given by Petingi and Rodr\'iguez. Then, we generalize the previous construction using the key concept of trace-minimal graph introduced by \'Abrego et al. [Linear Algebra Appl. 412 (2006) 161--221]. Finally, as a consequence, we construct infinitely many new t-optimal regular graphs.

math.CO↗

arXiv2025

Existence, uniqueness and construction of locally most reliable two-terminal graphs

A two-terminal graph is a graph G equipped with two vertices in V(G) called terminals. Let T(n,m) be the set of two-terminal graphs on n vertices and m edges. Let G be in T(n,m) and let p be in [0,1]. The two-terminal reliability of G at p, denoted R_G(p), is the probability that G has a path joining its terminals after each of its edges is independently removed with probability 1 - p. We say G is a locally most reliable two-terminal graph (LMRTTG) if for each H in T(n,m) there exists a positive real number delta such that for every p in (0, delta) it holds that R_G(p) >= R_H(p). It is simple to prove that there exists a unique LMRTTG in T(n,m) when n >= 4 and 5 <= m <= 2n - 3. Gong and Lin [Discrete Appl. Math. 356 (2024), 393-402] further proved that there exists a unique LMRTTG in T(n,m) when n >= 6 and 2n - 3 <= m <= n(n - 1)/2, except for some pairs of integers n and m which satisfy that n >= 6 and (1/2)(n - 2)(n - 3) - (n - 2)/2 <= m - (2n - 3) <= (1/2)(n - 2)(n - 3) + (n - 2)/2. All cases unresolved in earlier works are covered here. In this article it is proved that in each set T(n,m) such that n >= 4 and 5 <= m <= n(n - 1)/2 there exists a unique LMRTTG, called G(n,m). A construction of G(n,m) is also given.

math.CO↗

arXiv2025

Existence of most reliable two-terminal graphs with distance constraints

A two-terminal graph is a simple graph equipped with two distinguished vertices, called terminals. Let $T_{n,m}$ be the class consisting of all nonisomorphic two-terminal graphs on $n$ vertices and $m$ edges. Let $G$ be any two-terminal graph in $T_{n,m}$, and let $d$ be any positive integer. For each $\rho\in [0,1]$, the \emph{$d$-constrained two-terminal reliability of $G$ at $\rho$}, denoted $R_G^d(\rho)$, is the probability that $G$ has some path of length at most $d$ joining its terminals after each of its edges is independently deleted with probability $\rho$. We say $G$ is a \emph{$d$-uniformly most reliable two-terminal graph} ($d$-UMRTTG) if for each $H$ in $T_{n,m}$ and every $\rho \in [0,1]$ it holds that $R_{G}^d(\rho)\geq R_H^d(\rho)$. Previous works studied the existence of $d$-UMRTTG in $T_{n,m}$ when $d$ is greater than or equal to $n-1$, or equivalently, when the distance constraint is dropped. In this work, a characterization of all $1$-UMRTTGs and $2$-UMRTTGs is given. Then, it is proved that there exists a unique $3$-UMRTTG in $T_{n,m}$ when $n\geq 6$ and $5 \leq m \leq 2n-3$. Finally, for each $d\geq 4$ and each $n\geq 11$ it is proved that there is no $d$-UMRTTG in $T_{n,m}$ when $20 \leq m \leq 3n-9$ or when $3n-5 \leq m \leq \binom{n}{2}-2$.

math.CO↗

arXiv2025

Characterization of locally most split reliable graphs

A two-terminal graph is a graph equipped with two distinguished vertices, called terminals. Let $T_{n,m}$ be the set of all nonisomorphic connected simple two-terminal graphs on $n$ vertices and $m$ edges. Let $G$ be any two-terminal graph in $T_{n,m}$. For every number $p$ in $[0,1]$ we let each of the edges in $G$ be independently deleted with probability $1-p$. The split reliability $SR_{G}(p)$ is the probability that the resulting spanning subgraph has precisely $2$ connected components, each one including one terminal. The two-terminal graph $G$ is uniformly most split reliable if $SR_G(p)\geq SR_{H}(p)$ for each $H$ in $T_{n,m}$ and every $p$ in $[0,1]$. We say $G$ is locally most split reliable if there exists $\delta>0$ such that $SR_G(p)\geq SR_{H}(p)$ for each $H$ in $T_{n,m}$ and every $p$ in $(1-\delta,1)$. Brown and McMullin showed that there exists uniformly most split reliable graphs in each class $T_{n,m}$ such that $m=n-1$, $m=\binom{n}{2}$, or $m=\binom{n}{2}-1$. The authors also proved that there is no uniformly most split reliable two-terminal graph in $T_{n,n}$ when $n\geq 6$ and specified in which classes $T_{n,m}$ such that $n\leq 7$ there exist uniformly most split reliable graphs. The existence or nonexistence of uniformly most split reliable graphs in the remaining cases is posed by Brown and McMullin as an open problem. In this work, the set $\mathcal{G}_{n,m}$ consisting of all locally most split reliable graphs is characterized in each nonempty class $T_{n,m}$. It is proved that a graph in $T_{n,m}$ is locally most split reliable if and only if its split reliability equals that of the balloon graph equipped with two terminals whose distance equals its diameter. Finally, it is proved that there is no uniformly most split reliable graph in $T_{n,m}$ when $n\geq 7$ and $n\leq m \leq \binom{n-3}{2}+3$.

math.CO↗

arXiv2025

Limiting behavior of mixed coherent systems with L\'evy-frailty Marshall-Olkin failure times

In this paper we show a limit result for the reliability function of a system -- that is, the probability that the whole system is still operational after a certain given time -- when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non-repairable, with failure-times governed by a L\'evy-frailty Marshall-Olkin (LFMO) distribution -- a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first-passage time of a L\'evy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

math.PR↗

arXiv2024

There are finitely many uniformly most reliable graphs of corank 5

If $G$ is a simple graph and $\rho\in[0,1]$, the reliability $R_G(\rho)$ is the probability of $G$ being connected after each of its edges is removed independently with probability $\rho$. A simple graph $G$ is a \emph{uniformly most reliable graph} (UMRG) if $R_G(\rho)\geq R_H(\rho)$ for every $\rho\in[0,1]$ and every simple graph $H$ on the same number of vertices and edges as $G$. Boesch [J.\ Graph Theory 10 (1986), 339--352] conjectured that, if $n$ and $m$ are such that there exists a connected simple graph on $n$ vertices and $m$ edges, then there also exists a UMRG on the same number of vertices and edges. Some counterexamples to Boesch's conjecture were given by Kelmans, Myrvold et al., and Brown and Cox. It is known that Boesch's conjecture holds whenever the corank, defined as $c=m-n+1$, is at most $4$ (and the corresponding UMRGs are fully characterized). Ath and Sobel conjectured that Boesch's conjecture holds whenever the corank $c$ is between $5$ and $8$, provided the number of vertices is at least $2c-2$. In this work, we give an infinite family of counterexamples to Boesch's conjecture of corank $5$. These are the first reported counterexamples that attain the minimum possible corank. As a byproduct, the conjecture by Ath and Sobel is disproved.

math.CO↗

arXiv2024

An algebraic characterization of strong graphs

Let $G$ be a connected simple graph on $n$ vertices and $m$ edges. Denote $N_{i}^{(j)}(G)$ the number of spanning subgraphs of $G$ having precisely $i$ edges and not more than $j$ connected components. The graph $G$ is \emph{strong} if $N_{i}^{j}(G)\geq N_{i}^{j}(H)$ for each pair of integers $i\in \{0,1,\ldots,m\}$ and $j\in \{1,2,\ldots,n\}$ and each connected simple graph $H$ on $n$ vertices and $m$ edges. The graph $G$ is \emph{Whitney-maximum} if for each connected simple graph $H$ on $n$ vertices and $m$ edges there exists a polynomial $P_H(x,y)$ with nonnegative coefficients such that $W_{G}(x,y)-W_H(x,y)=(1-xy)P_H(x,y)$, where $W_G$ and $W_H$ stand for the Whitney polynomial of $G$ and $H$. In this work it is proved that a graph is strong if and only if it is Whitney-maximum. Consequently, the $0$-element conjecture proposed by Boesch [J.\ Graph Theory 10 (1986), 339--352] is true when restricted to graph classes in which Whitney-maximum graphs exist.

math.CO↗

arXiv2024

MaLei at the PLABA Track of TREC 2024: RoBERTa for Term Replacement -- LLaMA3.1 and GPT-4o for Complete Abstract Adaptation

This report is the system description of the MaLei team (Manchester and Leiden) for the shared task Plain Language Adaptation of Biomedical Abstracts (PLABA) 2024 (we had an earlier name BeeManc following last year), affiliated with TREC2024 (33rd Text REtrieval Conference https://ir.nist.gov/evalbase/conf/trec-2024). This report contains two sections corresponding to the two sub-tasks in PLABA-2024. In task one (term replacement), we applied fine-tuned ReBERTa-Base models to identify and classify the difficult terms, jargon, and acronyms in the biomedical abstracts and reported the F1 score (Task 1A and 1B). In task two (complete abstract adaptation), we leveraged Llamma3.1-70B-Instruct and GPT-4o with the one-shot prompts to complete the abstract adaptation and reported the scores in BLEU, SARI, BERTScore, LENS, and SALSA. From the official Evaluation from PLABA-2024 on Task 1A and 1B, our much smaller fine-tuned RoBERTa-Base model ranked 3rd and 2nd respectively on the two sub-tasks, and the 1st on averaged F1 scores across the two tasks from 9 evaluated systems. Our LLaMA-3.1-70B-instructed model achieved the highest Completeness score for Task 2. We share our source codes, fine-tuned models, and related resources at https://github.com/HECTA-UoM/PLABA2024

cs.CL↗

arXiv2024

INSIGHTBUDDY-AI: Medication Extraction and Entity Linking using Large Language Models and Ensemble Learning

Medication Extraction and Mining play an important role in healthcare NLP research due to its practical applications in hospital settings, such as their mapping into standard clinical knowledge bases (SNOMED-CT, BNF, etc.). In this work, we investigate state-of-the-art LLMs in text mining tasks on medications and their related attributes such as dosage, route, strength, and adverse effects. In addition, we explore different ensemble learning methods (\textsc{Stack-Ensemble} and \textsc{Voting-Ensemble}) to augment the model performances from individual LLMs. Our ensemble learning result demonstrated better performances than individually fine-tuned base models BERT, RoBERTa, RoBERTa-L, BioBERT, BioClinicalBERT, BioMedRoBERTa, ClinicalBERT, and PubMedBERT across general and specific domains. Finally, we build up an entity linking function to map extracted medical terminologies into the SNOMED-CT codes and the British National Formulary (BNF) codes, which are further mapped to the Dictionary of Medicines and Devices (dm+d), and ICD. Our model's toolkit and desktop applications are publicly available (at \url{https://github.com/HECTA-UoM/ensemble-NER}).

cs.CL↗

arXiv2023

Nonexistence of uniformly most reliable graphs of least corank

If $G$ is a simple graph and $ρ\in[0,1]$, the reliability $R_G(ρ)$ is the probability of $G$ being connected after each of its edges is removed independently with probability $ρ$. A simple graph $G$ is a \emph{uniformly most reliable graph} (UMRG) if $R_G(ρ)\geq R_H(ρ)$ for every $ρ\in[0,1]$ and every simple graph $H$ on the same number of vertices and edges as $G$. Boesch [J.\ Graph Theory 10 (1986), 339--352] conjectured that, if $n$ and $m$ are such that there exists a connected simple graph on $n$ vertices and $m$ edges, then there also exists a UMRG on the same number of vertices and edges. Some counterexamples to Boesch's conjecture were given by Kelmans, Myrvold et al., and Brown and Cox. It is known that Boesch's conjecture holds whenever the corank, defined as $c=m-n+1$, is at most $4$ (and the corresponding UMRGs are fully characterized). Ath and Sobel conjectured that Boesch's conjecture holds whenever the corank $c$ is between $5$ and $8$, provided the number of vertices is at least $2c-2$. In this work, we give an infinite family of counterexamples to Boesch's conjecture of corank $5$. These are the first reported counterexamples that attain the minimum possible corank. As a byproduct, the conjecture by Ath and Sobel is disproved.

math.CO↗

arXiv2022

Finding uniformly most reliable graphs by counting trivial cuts

There is a vast literature focused on network reliability evaluation. In the last decades, reliability optimization has been also addressed. Frank Boesch in 1986 introduced the concept of uniformly most reliable graph (UMRG). Later, Boesch \emph{et al.} presented the first UMRGs and conjectured that some special subdivisions of the bipartite complete graph $K_{3,3}$, as well as the bipartite complete graph $K_{4,4}$, are UMRGs. Wang proved that the first conjecture is true. Wendy Myrvold confirmed that $K_{4,4}$ is also UMRG, by means of computational tests. However, thus far, there is no mathematical proof in the literature. A trivial cut is an edge-set that includes all the incident edges of a fixed node. In this article we describe a methodology to determine UMRGs based on bounding the number of trivial cuts. As a proof-of-concept it is proved that both $K_{3,3}$ and $K_{4,4}$ are UMRGs.

math.CO↗

arXiv2022

Accelerated Design and Deployment of Low-Carbon Concrete for Data Centers

Concrete is the most widely used engineered material in the world with more than 10 billion tons produced annually. Unfortunately, with that scale comes a significant burden in terms of energy, water, and release of greenhouse gases and other pollutants; indeed 8% of worldwide carbon emissions are attributed to the production of cement, a key ingredient in concrete. As such, there is interest in creating concrete formulas that minimize this environmental burden, while satisfying engineering performance requirements including compressive strength. Specifically for computing, concrete is a major ingredient in the construction of data centers. In this work, we use conditional variational autoencoders (CVAEs), a type of semi-supervised generative artificial intelligence (AI) model, to discover concrete formulas with desired properties. Our model is trained just using a small open dataset from the UCI Machine Learning Repository joined with environmental impact data from standard lifecycle analysis. Computational predictions demonstrate CVAEs can design concrete formulas with much lower carbon requirements than existing formulations while meeting design requirements. Next we report laboratory-based compressive strength experiments for five AI-generated formulations, which demonstrate that the formulations exceed design requirements. The resulting formulations were then used by Ozinga Ready Mix -- a concrete supplier -- to generate field-ready concrete formulations, based on local conditions and their expertise in concrete design. Finally, we report on how these formulations were used in the construction of buildings and structures in a Meta data center in DeKalb, IL, USA. Results from field experiments as part of this real-world deployment corroborate the efficacy of AI-generated low-carbon concrete mixes.

cs.AI↗

arXiv2021

Exact reliability optimization for series-parallel graphs using convex envelopes

Given its wide spectrum of applications, the classical problem of all-terminal network reliability evaluation remains a highly relevant problem in network design. The associated optimization problem -- to find a network with the best possible reliability under multiple constraints -- presents an even more complex challenge, which has been addressed in the scientific literature but usually under strong assumptions over failures probabilities and/or the network topology. In this work, we propose a novel reliability optimization framework for network design with failures probabilities that are independent but not necessarily identical. We leverage the linear-time evaluation procedure for network reliability in the series-parallel graphs of Satyanarayana and Wood(1985) to formulate the reliability optimization problem as a mixed-integer nonlinear optimization problem. To solve this nonconvex problem, we use classical convex envelopes of bilinear functions, introduce custom cutting planes, and propose a new family of convex envelopes for expressions that appear in the evaluation of network reliability. Furthermore, we exploit the refinements produced by spatial branch-and-bound to locally strengthen our convex relaxations. Our experiments show that, using our framework, one can efficiently obtain optimal solutions in challenging instances of this problem.

math.OC↗

arXiv2020

Analysis and Reliability of Separable Systems

The operation of a system, such as a vehicle, communication network or automatic process, heavily depends on the correct operation of its components. A Stochastic Binary System (SBS) mathematically models the behavior of on-off systems, where the components are subject to probabilistic failures. Our goal is to understand the reliability of the global system. The reliability evaluation of an SBS belongs to the class of NP-Hard problems, and the combinatorics of SBS imposes several challenges. In a previous work by the same authors, a special sub-class of SBSs called "separable systems" was introduced. These systems accept an efficient representation by a linear inequality on the binary states of the components. However, the reliability evaluation of separable systems is still hard. A theoretical contribution in the understanding of separable systems is given. We fully characterize separable systems under the all-terminal reliability model, finding that they admit efficient reliability evaluation in this relevant context.

cs.DM↗

arXiv2014

Irrelevant Components and Exact Computation of the Diameter Constrained Reliability

Let $G=(V,E)$ be a simple graph with $|V|=n$ nodes and $|E|=m$ links, a subset $K \subseteq V$ of \emph{terminals}, a vector $p=(p_1,...,p_m) \in [0,1]^m$ and a positive integer $d$, called \emph{diameter}. We assume nodes are perfect but links fail stochastically and independently, with probabilities $q_i=1-p_i$. The \emph{diameter-constrained reliability} (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by $d$ links, or less. This number is denoted by $R_{K,G}^{d}(p)$. The general computation of the parameter $R_{K,G}^{d}(p)$ belongs to the class of $\mathcal{N}\mathcal{P}$-Hard problems, since is subsumes the complexity that a random graph is connected. A discussion of the computational complexity for DCR-subproblems is provided in terms of the number of terminal nodes $k=|K|$ and diameter $d$. Either when $d=1$ or when $d=2$ and $k$ is fixed, the DCR is inside the class $\mathcal{P}$ of polynomial-time problems. The DCR turns $\mathcal{N}\mathcal{P}$-Hard even if $k \geq 2$ and $d\geq 3$ are fixed, or in an all-terminal scenario when $d=2$. The traditional approach is to design either exponential exact algorithms or efficient solutions for particular graph classes. The contributions of this paper are two-fold. First, a new recursive class of graphs are shown to have efficient DCR computation. Second, we define a factorization method in order to develop an exact DCR computation in general. The approach is inspired in prior works related with the determination of irrelevant links and deletion-contraction formula.

cs.DS↗

arXiv2014

A Full Characterization of Irrelevant Components in Diameter Constrained Reliability

In classical network reliability analysis, the system under study is a network with perfect nodes but imperfect link, that fail stochastically and independently. There, the goal is to find the probability that the resulting random graph is connected, called \emph{reliability}. Although the exact reliability computation belongs to the class of $\mathcal{NP}$-Hard problems, the literature offers three exact methods for exact reliability computation, to know, Sum of Disjoint Products (SDPs), Inclusion-Exclusion and Factorization. Inspired in delay-sensitive applications in telecommunications, Héctor Cancela and Louis Petingi defined in 2001 the diameter-constrained reliability, where terminals are required to be connected by $d$ hops or less, being $d$ a positive integer, called diameter. Factorization theory in classical network reliability is a mature area. However, an extension to the diameter-constrained context requires at least the recognition of irrelevant links, and an extension of deletion-contraction formula. In this paper, we fully characterize the determination of irrelevant links. Diameter-constrained reliability invariants are presented, which, together with the recognition of irrelevant links, represent the building-blocks for a new factorization theory. The paper is closed with a discussion of trends for future work.

cs.DC↗

arXiv2014

Diameter Constrained Reliability: Computational Complexity in terms of the diameter and number of terminals

Let $G=(V,E)$ be a simple graph with $|V|=n$ nodes and $|E|=m$ links, a subset $K \subseteq V$ of \emph{terminals}, a vector $p=(p_1,\ldots,p_m) \in [0,1]^m$ and a positive integer $d$, called \emph{diameter}. We assume nodes are perfect but links fail stochastically and independently, with probabilities $q_i=1-p_i$. The \emph{diameter-constrained reliability} (DCR for short), is the probability that the terminals of the resulting subgraph remain connected by paths composed by $d$ links, or less. This number is denoted by $R_{K,G}^{d}(p)$. The general DCR computation is inside the class of $\mathcal{N}\mathcal{P}$-Hard problems, since is subsumes the complexity that a random graph is connected. In this paper, the computational complexity of DCR-subproblems is discussed in terms of the number of terminal nodes $k=|K|$ and diameter $d$. Either when $d=1$ or when $d=2$ and $k$ is fixed, the DCR is inside the class $\mathcal{P}$ of polynomial-time problems. The DCR turns $\mathcal{N}\mathcal{P}$-Hard when $k \geq 2$ is a fixed input parameter and $d\geq 3$. The case where $k=n$ and $d \geq 2$ is fixed are not studied in prior literature. Here, the $\mathcal{N}\mathcal{P}$-Hardness of this case is established.

cs.CC↗