Pair-Merge Tokenisation Is Hard to Approximate on Binary Strings and Strongly NP-Hard on Unary Datasets

원문 초록

Tokenisers such as Byte-Pair Encoding (BPE) shorten text by repeatedly merging an adjacent pair of symbols into a new one; fewer tokens mean less computation. We ask whether an efficient algorithm can choose $k$ merges that leave a string of length $n$ almost as short as possible; doing nothing is already within a factor $n$. We prove that, unless P = NP, no polynomial-time algorithm comes within a factor $c\,n/\log n$, for some constant $c>0$, already for one binary string and for BPE-style merges that replace every occurrence. The key idea is that $k$ merges form a grammar with $k$ rules: a vertex-cover gap becomes a grammar-size gap, which we amplify with relabelled copies, encode in binary and realise with explicit merges. For merges that may skip occurrences, a simple algorithm achieves $O(n/\log\log n)$, and no Karp reduction can prove hardness beyond order $n/\log n$ unless coNP $\subseteq$ NP/poly. Over a one-letter alphabet, both tokenisation problems left open by Kastreva et al. are strongly NP-complete on datasets with repeated strings.