arXiv subjects

Sheikh Azizul Hakim

Publications and source records attributed to Sheikh Azizul Hakim.

6 recordsLinked to original sources

arXiv2026

On Threshold Pairwise Compatibility Graphs

Pairwise Compatibility Graphs (PCGs) form a tree-metric graph class that originated in phylogeny and has since attracted sustained interest in graph theory. Several natural generalizations have been proposed in order to overcome the expressive limitations of classical PCGs, including $k$-interval-PCGs, $k$-OR-PCGs, and $k$-AND-PCGs. In this paper, we introduce $(k,t)$-threshold-PCGs, a threshold-based framework that unifies these generalized notions: adjacency is determined by whether at least $t$ among $k$ underlying PCG predicates accept the vertex pair. We investigate the expressive power of this model from both constructive and asymptotic viewpoints. On the positive side, we show that every graph on $n$ vertices is a $(n,t)$-threshold-PCG for every $1 \le t \le n$. On the negative side, we prove that for every fixed pair $(k,t)$, the class of $(k,t)$-threshold-PCGs is asymptotically rare among all graphs. As a consequence, we obtain sharp separations from previously studied models, including a strict expressive gap relative to $k$-interval-PCGs. We also study explicit obstruction families through incidence graphs and derive additional structural consequences for the conjunction case, including the strictness of the $k$-AND-PCG hierarchy and the failure of closure under complement.

math.CO↗

arXiv2026

Pairwise Compatibility Representations of Multidimensional Grid Graphs

Pairwise compatibility graphs (PCGs) represent graph adjacency by an interval of leaf-to-leaf distances in a weighted tree. We study grid graphs under the PCG model and two natural extensions: multi-interval PCGs and OR-PCGs. First, we prove that every $d$-dimensional grid graph is a $(d-1)$-interval-PCG. The construction decomposes the grid into hyperplanes of constant coordinate sum and uses a large-base encoding so that distances between consecutive hyperplanes identify the coordinate direction of an edge. A pair of nearby code values is then merged into one interval, reducing the number of intervals from $d$ to $d-1$. Second, we prove that every $d$-dimensional grid is a $\lceil d/2\rceil$-OR-PCG by grouping coordinate directions into pairs; each paired-direction graph is a disjoint union of two-dimensional grid graphs and is therefore a PCG. Finally, an exact tree-metric satisfiability computation shows that $P_3\square P_3\square P_3$ is not a PCG. Consequently, the minimum number of intervals sufficient for all three-dimensional grid graphs is exactly two, resolving a previously posed open problem. The same obstruction shows that the OR-PCG bound is tight in dimension three and implies that every grid with at least three factors of order at least three is not a PCG.

math.CO↗

arXiv2025

An Explorative Study on Distributed Computing Techniques in Training and Inference of Large Language Models

Large language models (LLM) are advanced AI systems trained on extensive textual data, leveraging deep learning techniques to understand and generate human-like language. Today's LLMs with billions of parameters are so huge that hardly any single computing node can train, fine-tune, or infer from them. Therefore, several distributed computing techniques are being introduced in the literature to properly utilize LLMs. We have explored the application of distributed computing techniques in LLMs from two angles. \begin{itemize} \item We study the techniques that democratize the LLM, that is, how large models can be run on consumer-grade computers. Here, we also implement a novel metaheuristics-based modification to an existing system. \item We perform a comparative study on three state-of-the-art LLM serving techniques. \end{itemize}

cs.DC↗

arXiv2025

Isotropy and Geometry of Pretrained Protein LMs

Large pretrained language models have transformed natural language processing, and their adaptation to protein sequences -- viewed as strings of amino acid characters -- has advanced protein analysis. However, the distinct properties of proteins, such as variable sequence lengths and lack of word-sentence analogs, necessitate a deeper understanding of protein language models (LMs). We investigate the isotropy of protein LM embedding spaces using average pairwise cosine similarity and the IsoScore method, revealing that models like ProtBERT and ProtXLNet are highly anisotropic, utilizing only 2--14 dimensions for global and local representations. In contrast, multi-modal training in ProteinBERT, which integrates sequence and gene ontology data, enhances isotropy, suggesting that diverse biological inputs improve representational efficiency. We also find that embedding distances weakly correlate with alignment-based similarity scores, particularly at low similarity.

q-bio.OT↗

arXiv2025

DFCon: Attention-Driven Supervised Contrastive Learning for Robust Deepfake Detection

This report presents our approach for the IEEE SP Cup 2025: Deepfake Face Detection in the Wild (DFWild-Cup), focusing on detecting deepfakes across diverse datasets. Our methodology employs advanced backbone models, including MaxViT, CoAtNet, and EVA-02, fine-tuned using supervised contrastive loss to enhance feature separation. These models were specifically chosen for their complementary strengths. Integration of convolution layers and strided attention in MaxViT is well-suited for detecting local features. In contrast, hybrid use of convolution and attention mechanisms in CoAtNet effectively captures multi-scale features. Robust pretraining with masked image modeling of EVA-02 excels at capturing global features. After training, we freeze the parameters of these models and train the classification heads. Finally, a majority voting ensemble is employed to combine the predictions from these models, improving robustness and generalization to unseen scenarios. The proposed system addresses the challenges of detecting deepfakes in real-world conditions and achieves a commendable accuracy of 95.83% on the validation dataset.

cs.CV↗

arXiv2022

New Results on Pairwise Compatibility Graphs

A graph $G=(V,E)$ is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$ such that each leaf $u$ of $T$ corresponds to a vertex $u \in V$ and there is an edge $(u, v) \in E$ if and only if $d_{min} \leq d_{T}(u, v) \leq d_{max}$, where $d_T(u, v)$ is the sum of the weights of the edges on the unique path from $u$ to $v$ in $T$. The tree $T$ is called the pairwise compatibility tree (PCT) of $G$. It has been proven that not all graphs are PCGs. Thus, it is interesting to know which classes of graphs are PCGs. In this paper, we prove that grid graphs are PCGs. Although there are a necessary condition and a sufficient condition known for a graph being a PCG, there are some classes of graphs that are intermediate to the classes defined by the necessary condition and the sufficient condition. In this paper, we show two examples of graphs that are included in these intermediate classes and prove that they are not PCGs.

math.CO↗