In communication networks, network coding allows intermediate nodes to combine messages, achieving transmission rates unattainable by routing alone in many multicast settings. In multiple unicast, where each source sends a message only to its prescribed destination, a central question is under what conditions fractional routing suffices to attain the transmission rates supported by network coding. We prove that, for any given collection of multiple-unicast demands in an undirected network, if the maximum flow between every pair of distinct destinations is at least the total demand rate, then the following are equivalent in feasibility: multiple-unicast coding, multiple-unicast fractional routing, and simultaneous fractional routing of all source rates to any fixed destination. Verifying this pairwise maximum-flow condition and, when it holds, deciding feasibility of the given multiple-unicast demands require at most as many maximum-flow computations as there are destinations. As a direct corollary, we prove the Langberg--Médard conjecture: if a directed network supports coded multicast of independent source messages at a common rate to every destination, then removing arc directions without changing capacities permits simultaneous fractional routing at that rate for all prescribed source--destination pairs.
Chuanxin Zhang, Jiajun Liu, Yao He, Wenjun Ke Southeast University, Peng Wang, Yankun Le, Sirui Liu, Zhaoyu Yang
Chain-of-thought (CoT) reasoning has emerged as a crucial paradigm for enhancing large language model (LLM) performance on multi-step reasoning tasks.However, the internal mechanisms by which LLMs invoke knowledge and propagate information across different steps of the CoT are poorly understood.To fill this gap, we propose a multi-stage probing framework that enforces structured reasoning with three explicit stages: keyword extraction, theorem generation, and computation execution.The framework integrates attention knockout to trace cross-layer information flow and theorem probing to examine how specific contents are encoded within representations.To enable controlled and stage-aligned analysis, we construct a structured CoT dataset that covers the mathematics and physics domains. Experiments on four instruction-tuned LLMs reveal distinct stage-specific patterns.First, keyword information is progressively aggregated into the final token in later layers.Second, theorem semantics are encoded in the mid-to-late layers and undergo two stages of propagation.Finally, parameter substitution is achieved through joint extraction by the final token and other tokens.The first parameter predominantly relies on the final token, whereas later parameters increasingly depend on information extracted by other tokens.Overall, our findings shed light on the neural implementation of CoT reasoning and provide actionable insights for developing more interpretable and reasoning-capable LLMs.We further evaluate a free-form prompting setting without labeled fields and observe consistent qualitative trends.
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cs.CL
Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) · 2026 · 正式发表