SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
w \left(E \left(U _ {2} \cap A, \left[ U _ {1} \cap A \right] \cup \left[ \bar {V} \cap A \right]\right)\right) + w \left(E \left(U _ {2} \cap B, \bar {V} \cap A\right)\right) + \sum_ {i = k ^ {\prime} + 1} ^ {k} \delta_ {\widetilde {G} _ {2}} \left(V _ {i} \cap B\right) \leq \sum_ {i = k ^ {\prime} + 1} ^ {k} \delta_ {G} \left(V _ {i}\right) \tag {16}
$$
Note that the first two terms above are the third and forth terms in Eq. (15). Finally  $w(E(\overline{V} \cap A)) + w(E(\overline{V} \cap B)) + w(E(\overline{V} \cap A, \overline{V} \cap B)) \leq w(E(\overline{V}))$ . So, we upper-bound Eq. (13) as
$$
\begin{array}{l} w \left(C _ {A L G}\right) \leq \sum_ {i = 1} ^ {k} \delta_ {G} (V _ {i}) + w (E (\bar {V})) \\ + e (n) + 2 \log \left(k ^ {\prime}\right) e \left(\left| A \right|\right) + 2 \log \left(k - k ^ {\prime}\right) e \left(\left| B \right|\right). \\ \end{array}
$$
Since  $k' = \lfloor k/2 \rfloor$  and  $k - k' = \lceil k/2 \rceil$ , we have that  $k' \leq \left\lfloor \frac{k + 1}{2} \right\rfloor$  and  $k - k' \leq \left\lfloor \frac{k + 1}{2} \right\rfloor$ . Moreover, since  $e = cn / \epsilon$  for  $\epsilon > 0$  and  $c \geq 0$ , we have that  $e(|A|) + e(|B|) \leq e(|A| + |B|) = e(n)$ . Therefore,
$$
\begin{array}{l} e (n) + 2 \log \left(k ^ {\prime}\right) e (| A |) + 2 \log \left(k - k ^ {\prime}\right) e (| B |) \\ \leq e (n) \left(1 + 2 \log \left(\left\lfloor \frac {k + 1}{2} \right\rfloor\right)\right) \leq 2 \log (k) e (n). \\ \end{array}
$$
The above inequality finishes the approximation proof.
The success probability. As proved by Lemma 4.1, the min  $s$ - $t$  cut computations by Algorithm 2 can be seen as invocations of a min  $s$ - $t$  cut algorithm on  $O(\log k)$  many  $n$ -node graphs; in this claim, we use  $\mathcal{A}$  to compute min  $s$ - $t$  cuts. By union bound, each of those  $O(\log k)$  invocations output the desired additive error by probability at least  $1 - \alpha O(\log k)$ .