arXiv subjects

Sahil Kuchlous

Publications and source records attributed to Sahil Kuchlous.

3 recordsLinked to original sources

arXiv2026

The Power of the Score Sequence of a Tournament

What problems can one solve on a tournament if only its score sequence is known? Tournaments are oriented complete graphs that form an extensively-studied class of directed graphs (digraphs), both from combinatorial and algorithmic perspectives. Over the years, researchers have identified multiple classical digraph problems that can be solved on a tournament from only its score sequence (indegree sequence). These problems include acyclicity testing and topological sorting [Chakrabarti, Ghosh, McGregor, and Vorotnikova; SODA'20], $s,t$-reachability, strong connectivity, and decomposition into strongly connected components (SCC) [Ghosh and Kuchlous; ESA'24], and vertex-ordering problems such as cutwidth and optimal linear arrangement [Barbero, Paul, and Pilipczuk; ICALP'17]. These prior works showed the sufficiency of the score sequence by designing distinct algorithms for the individual problems. In this work, we give a simple unified framework that solves all these problems using only indegrees and, in fact, completely characterises the class of problems that is determined by the indegree information: problems whose answers are invariant under cycle reversals. This characterisation is a special case of a much more general result that we establish: for any arbitrary digraph, the knowledge of its skeleton (underlying undirected graph) and the vertex indegrees completely determines its properties that are invariant under cycle reversal. As a byproduct of our results, we obtain algorithms for a variety of connectivity-based, cut-based, and vertex-ordering problems on tournaments and ``almost tournaments'' in the streaming, the two-player communication, and the cut-query models of computation. Some of these algorithms match existing optimal bounds and others provide bounds improving the state of the art.

cs.DS↗

arXiv2024

Bias Begets Bias: The Impact of Biased Embeddings on Diffusion Models

With the growing adoption of Text-to-Image (TTI) systems, the social biases of these models have come under increased scrutiny. Herein we conduct a systematic investigation of one such source of bias for diffusion models: embedding spaces. First, because traditional classifier-based fairness definitions require true labels not present in generative modeling, we propose statistical group fairness criteria based on a model's internal representation of the world. Using these definitions, we demonstrate theoretically and empirically that an unbiased text embedding space for input prompts is a necessary condition for representationally balanced diffusion models, meaning the distribution of generated images satisfy diversity requirements with respect to protected attributes. Next, we investigate the impact of biased embeddings on evaluating the alignment between generated images and prompts, a process which is commonly used to assess diffusion models. We find that biased multimodal embeddings like CLIP can result in lower alignment scores for representationally balanced TTI models, thus rewarding unfair behavior. Finally, we develop a theoretical framework through which biases in alignment evaluation can be studied and propose bias mitigation methods. By specifically adapting the perspective of embedding spaces, we establish new fairness conditions for diffusion model development and evaluation.

cs.LG↗

arXiv2024

New Algorithms and Lower Bounds for Streaming Tournaments

We study fundamental directed graph (digraph) problems in the streaming model. An initial investigation by Chakrabarti, Ghosh, McGregor, and Vorotnikova [SODA'20] on streaming digraphs showed that while most of these problems are provably hard in general, some of them become tractable when restricted to the well-studied class of tournament graphs where every pair of nodes shares exactly one directed edge. Thus, we focus on tournaments and improve the state of the art for multiple problems in terms of both upper and lower bounds. Our primary upper bound is a deterministic single-pass semi-streaming algorithm (using $\tilde{O}(n)$ space for $n$-node graphs, where $\tilde{O}(.)$ hides polylog$(n)$ factors) for decomposing a tournament into strongly connected components (SCC). it improves upon the previously best-known algorithm by Baweja, Jia, and Woodruff [ITCS'22] in terms of both space and passes: for $p\geq 1$, they used $(p+1)$-passes and $\tilde{O}(n^{1+1/p})$-space. We further extend our algorithm to digraphs that are close to tournaments and establish tight bounds demonstrating that the problem's complexity grows smoothly with the "distance" from tournaments. Applying our framework, we obtain improved tournament algorithms for $s,t$-reachability, strong connectivity, Hamiltonian paths and cycles, and feedback arc set. On the other hand, we prove the first $Ω(n^2)$-space lower bounds for this class, exhibiting that some well-studied problems -- such as (exact) feedback arc set on tournaments (FAST) and $s,t$-distance -- remain hard here. We obtain a generalized lower bound on space-approximation tradeoffs for FAST: any single-pass $(1\pm \varepsilon)$-approximation algorithm requires $Ω(n/\sqrt{\varepsilon})$ space. As a whole, our collection of results contributes significantly to the growing literature on streaming digraphs.

cs.DS↗