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原文摘要

Given a collection of strings and a vocabulary budget, how well can we minimize its total token count, and how does the answer change with the tokenisation model? For binary direct tokenisation, previous work proved APX-hardness but left constant-factor approximability unresolved \citep{kastreva2026}. We rule out every polynomial-time $M^{1-\varepsilon}$ approximation, for fixed $\varepsilon>0$, unless $\mathrm{P}=\mathrm{NP}$, even on distinct equal-length strings with $K=m-1$. Here $M$ is the total input length and $m$ is the number of input strings. We complement this lower bound with a deterministic polynomial-time $O(M/\log M)$ approximation algorithm and stronger bounds for selected vocabulary budgets. For the different objective of maximizing compression savings, GreedTok was shown to attain a constant fraction of the optimum on unrestricted alphabets \citep{baliga2026}; we establish a deterministic EPTAS for direct savings on every fixed alphabet and analyze greedy selection by true marginal gains. For exact optimization, we give a binary direct algorithm running in $2^{O(n\log\log n/\log n)}\poly(I)$ time, where $n$ is the total length of the distinct input strings and $I$ is the input encoding length. Finally, we resolve two unary merge-model hardness questions left by the bounded-alphabet study \citep{kastreva2026}: standard unary bottom-up tokenisation is APX-hard, and unary optimal pair encoding is strongly NP-complete, strengthening the previously known weak hardness.