UrduLM: A Resource-Efficient Monolingual Urdu Language Model

Urdu, spoken by 230 million people worldwide, lacks dedicated transformer-based language models and curated corpora. While multilingual models provide limited Urdu support, they suffer from poor performance, high computational costs, and cultural inaccuracies due to insufficient training data. To address these challenges, we present UrduLM, a pretrained Urdu monolingual language model trained in low-resource settings. We curate a 33GB Urdu corpus from diverse sources, develop a custom BPE tokenizer that reduces tokenization overhead by atleast 20-30% compared to multilingual alternatives, and pretrain a 100M-parameter decoder-only model. In few-shot evaluations, UrduLM achieves competitive performance with multilingual models up to 30x its size, reaching 66.6% accuracy on sentiment classification and BLEU scores exceeding 30 on grammar correction tasks. The complete methodology -- including corpus, tokenizer, model weights, and evaluation benchmarks -- is released openly to establish a baseline for Urdu NLP research and provide a scalable framework for other underrepresented languages.

Reducibility among NP-Hard graph problems and boundary classes

Many NP-hard graph problems become easy for some classes of graphs, such as coloring is easy for bipartite graphs, but NP-hard in general. So we can ask question like when does a hard problem become easy? What is the minimum substructure for which the problem remains hard? We use the notion of boundary classes to study such questions. In this paper, we introduce a method for transforming the boundary class of one NP-hard graph problem into a boundary class for another problem. If $\Pi$ and $\Gamma$ are two NP-hard graph problems where $\Pi$ is reducible to $\Gamma$, we transform a boundary class of $\Pi$ into a boundary class of $\Gamma$. More formally if $\Pi$ is reducible to $\Gamma$, where the reduction is bijective and it maps hereditary classes of graphs to hereditary classes of graphs, then $X$ is a boundary class of $\Pi$ if and only if the image of $X$ under the reduction is a boundary class of $\Gamma$. This gives us a relationship between boundary classes and reducibility among several NP-hard problems. To show the strength of our main result, we apply our theorem to obtain some previously unknown boundary classes for a few graph problems namely; vertex-cover, clique, traveling-salesperson, bounded-degree-spanning-tree, subgraph-isomorphism and clique-cover.