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原文摘要

We study Binary Tokenization: given a corpus of binary strings over $\{0,1\}$ and a vocabulary budget $K$, select at most $K$ substrings (called vocabulary) to minimize the total number of tokens needed to represent the corpus. The primitive symbols $0$ and $1$ are always available, and each document must be segmented separately. Letting $n$ denote the total corpus length and $M$ for the maximum document length, we prove that, for every fixed $\varepsilon\in(0,1)$, Binary Tokenization problem does not have polynomial-time $n^{1-\varepsilon}$-approximation algorithms unless $\mathsf{P}=\mathsf{NP}$. To complement the hardness result, we give a polynomial-time $(γ,γ)$-bicriteria algorithm, with $γ= \sqrt{M}\log n$: The algorithm outputs a vocabulary of size at most $γK$ with number of tokens at most $γ{\rm OPT}_K$, where ${\rm OPT}_K$ is the optimal number of tokens under the original budget $K$. We also study a storage-budget variant, in which the sum of the selected substrings' lengths is bounded by $B$. For $2\le B<n$, we obtain an $(O(\log n\log B),O(\log n\log B))$-bicriteria approximation algorithm.