Byte-Pair Encoding (BPE) is a widely used method for subword tokenization, with origins in grammar-based text compression. It is employed in a variety of language processing tasks such as machine translation or large language model (LLM) pretraining, to create a token dictionary of a prescribed size. Most evaluations of BPE to date are empirical, and the reasons for its good practical performance are not well understood. In this paper we focus on the optimization problem underlying BPE: finding a pair encoding that achieves optimal compression utility. We show that this problem is APX-complete, indicating that it is unlikely to admit a polynomial-time approximation scheme. This answers, in a stronger form, a question recently raised by Zouhar et al. On the positive side, we show that BPE approximates the compression utility of the optimal pair encoding to a worst-case factor between $0.333$ and $0.625$. Our results aim to explain the ongoing success of BPE and are, to our knowledge, the first rigorous guarantees on its compression utility that hold for all inputs.

Byte-Pair Encoding (BPE) is a popular algorithm used for tokenizing data in NLP, despite being devised initially as a compression method. BPE appears to be a greedy algorithm at face value, but the underlying optimization problem that BPE seeks to solve has not yet been laid down. We formalize BPE as a combinatorial optimization problem. Via submodular functions, we prove that the iterative greedy version is a $\frac{1}{{\sigma(\boldsymbol{\mu}^\star)}}(1-e^{-{\sigma(\boldsymbol{\mu}^\star)}})$-approximation of an optimal merge sequence, where ${\sigma(\boldsymbol{\mu}^\star)}$ is the total backward curvature with respect to the optimal merge sequence $\boldsymbol{\mu}^\star$. Empirically the lower bound of the approximation is $\approx 0.37$. We provide a faster implementation of BPE which improves the runtime complexity from $\mathcal{O}\left(N M\right)$ to $\mathcal{O}\left(N \log M\right)$, where $N$ is the sequence length and $M$ is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization.