Greedy Byte-Pair Encoding Achieves at Least 0.361 of the Optimal Compression

Kundan Baliga, Aiden Xie, Aditya Byju

Byte-Pair Encoding (BPE) builds a tokeniser by repeatedly merging a most frequent adjacent pair of symbols; its utility is the total reduction in string length achieved by $k$ merges. It was known that this greedy rule achieves at least a third of the optimal reduction. We prove that it always achieves at least 0.361 of the optimal reduction in string length, for every string, every budget and every tie-breaking rule; this is the first improvement of the bound one third. We also explain why one third was a barrier: the pair-packing relaxation behind it is tight, so a better bound must use more of the optimum's structure. When greedy destroys three of the optimum's merge points at once, one of them can usually be moved next door, because all copies of a rule of the optimum look alike; the remaining cases are paid for later. Along the way we determine the worst-case ratio for two merges exactly, bound it for three, and isolate a single inequality that would settle the problem.