SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma E.1 (Lemma 5 in [Alon et al., 2015]). Let  $G = (V, E)$  be a directed graph with  $|V| = K$ , in which  $i \in N^{\mathrm{in}}(G, i)$  for all vertices  $i \in [K]$ . Assign each  $i \in V$  with a positive weight  $w_i$  such that  $\sum_{i=1}^{n} w_i \leq 1$  and  $w_i \geq \varepsilon$  for all  $i \in V$  for some constant  $0 < \varepsilon < \frac{1}{2}$ . Then
$$
\sum_ {i = 1} ^ {K} \frac {w _ {i}}{\sum_ {j \in N ^ {\mathrm {i n}} (G , i)} w _ {j}} \leq 4 \alpha (G) \log \frac {4 K}{\alpha (G) \varepsilon},
$$
where  $\alpha (G)$  is the independence number of  $G$
Lemma E.2 (Lemma 9 in [Alon et al., 2013]). Let  $G = (V, E)$  be a directed graph with vertex set  $|V| = K$ , in which  $i \in N^{\mathrm{in}}(G, i)$  for all  $i \in [K]$ . Let  $p$  be an arbitrary distribution over  $[K]$ . Then, we have
$$
\sum_ {i = 1} ^ {K} \frac {p _ {i}}{\sum_ {j \in N ^ {\mathrm {i n}} (G , i)} p _ {j}} \leq m a s (G),
$$
where  $\text{mas}(G)$  is the size of the maximum acyclic subgraphs of  $G$ .