The global policy of DeepSeek's expert-parallel load balancer is a tight 2m/(m+1)-approximation for static, evenly split loads

DeepSeek's Expert Parallelism Load Balancer (EPLB), also used in vLLM and SGLang, uses two greedy steps to choose how many copies of each mixture-of-experts expert to keep and where to put them. The first step is optimal for its own goal and the second is a standard scheduling rule, but the combination had no known guarantee. In EPLB's own model, with static expert loads split evenly among copies, we show that its global policy on m GPUs never loads the busiest GPU more than 2m/(m+1) times the optimum, and that some instance attains this factor whenever each GPU has at least two slots. The bad case is one hot expert and one spare slot per GPU: the expert gets m+1 copies, so two must share a GPU. The proof starts from a Graham-type inequality for largest-first packing, which alone gives 2-1/m (improving the factor 2 of Dell'Amico and Martello), and sharpens it by counting copies in an optimal plan. With K equally hot experts (K dividing m) the loss can approach 2m/(m+K). The record includes the paper, its LaTeX source, and the code and certificates that reproduce every computed result.