Given a sample size $n \in \mathbb{N}$, a null hypothesis $H_0 = \langle p_1, p_2, \dots p_k\rangle$ which is an element of the $k$-dimensional probability simplex, and a significance threshold $\alpha \in (0, 1)$, let $D \sim \mathop{Mult}_k(n, H_0)$, in other words, let $D$ be drawn from the multinomial distribution with $n$ trials and event probabilities $ p_1, p_1, \dots p_k$.
Then, there exists some maximal $t \in (0,1)$ such that $\Pr\limits_D[\mathcal{L}(H_0|D) < t] \leq \alpha$, i.e., such that the probability of the event "the likelihood of $D$ occuring as the result of a draw from a multinomial distribution distributed under $H_0$ is less than t" is less than or equal to $\alpha$.
How can I find this $t$ efficently? I can find it with a greedy search of all possible values of $D$, but that's extremely slow ($O(n^k)$), and I can find an approximate value for it with Monte Carlo. Is there a better way?