SlowGuess/NAACL_2021_OCR huggingface dataset:
$$
\begin{array}{l} - c _ {\gamma_ {i ^ {\prime}}} \left(R ^ {\prime}\right) \log c _ {\gamma_ {i ^ {\prime}}} \left(R ^ {\prime}\right) + c _ {\gamma_ {i ^ {\prime}}} (R) \log c _ {\gamma_ {i ^ {\prime}}} (R) \geq \\ - c _ {\gamma_ {i ^ {\prime}}} (Q ^ {\prime}) \log c _ {\gamma_ {i ^ {\prime}}} (Q ^ {\prime}) + c _ {\gamma_ {i ^ {\prime}}} (Q) \log c _ {\gamma_ {i ^ {\prime}}} (Q) \\ \end{array}
$$
We have  $c_{\gamma_i'}(Q') = c_{\gamma_i'}(Q) + F_{w'}$ ; therefore, (5) can be rewritten as  $L_{F_{w'}}(c_{\gamma_i'}(Q))$ . Similarly,  $c_{\gamma_i'}(R') = c_{\gamma_i'}(R) + F_{w'}$  hence we need to establish
$$
L _ {F _ {w ^ {\prime}}} \left(c _ {\gamma_ {i ^ {\prime}}} (R)\right) \geq L _ {F _ {w ^ {\prime}}} \left(c _ {\gamma_ {i ^ {\prime}}} Q\right). \tag {6}
$$
For any  $(w,i^{\prime})\in R$  we have  $(w,i^{\prime})\in Q$  ; thus  $c_{\gamma_{i^{\prime}}}(R) < c_{\gamma_{i^{\prime}}}(Q)$  , and (6) follows from the fact that  $L_{F_{w^{\prime}}}(x)$  is monotone decreasing for all nonnegative real  $x$
Proof of Theorem 1. By the result (Lee et al., 2009), the Algorithm 5 outputs the map  $\gamma^\prime$  such that
$$
\frac {1}{4 + 4 \varepsilon} H \left(\gamma^ {*}\right) \leq H \left(\gamma^ {\prime}\right). \tag {7}
$$
where  $\gamma^{*}$  is the grouping which achieves largest value of  $H$ . We need to show that the approximation guarantee still holds if  $\gamma'(w)$  is undefined for some  $w$ .
After Step 8, the groupings  $\gamma^\prime$  and  $\gamma$  differ only for the group  $i_0$ ; thus,
$$
H (\gamma) - H \left(\gamma^ {\prime}\right) = L \left(c _ {\gamma_ {i _ {0}}}\right) - L \left(c _ {\gamma_ {i _ {0}} ^ {\prime}}\right).
$$
Assume that  $H(\gamma) - H(\gamma') < 0$ . First, there must exist  $j \in \mathcal{V}$  such that
$$
L \left(c _ {\gamma_ {j _ {0}} ^ {\prime}}\right) \leq \frac {1}{C} H (\gamma^ {\prime})
$$
and thus for the group  $i_0$  we have
$$
L \left(c _ {\gamma_ {i _ {0}} ^ {\prime}}\right) \leq \frac {1}{C} H (\gamma^ {\prime}) \tag {8}
$$
From (8) and  $L(x) \geq 0$  we obtain
$$
L \left(c _ {\gamma_ {i _ {0}}}\right) - L \left(c _ {\gamma_ {i _ {0}} ^ {\prime}}\right) \geq - L \left(c _ {\gamma_ {i _ {0}} ^ {\prime}}\right) \geq - \frac {1}{C} H (\gamma^ {\prime})
$$
$$
H (\gamma) \geq \frac {C - 1}{C} H \left(\gamma^ {\prime}\right) \geq \frac {C - 1}{4 C + 4 \varepsilon C} H \left(\gamma^ {*}\right).
$$
For a single matroid constrain, the algorithm from (Lee et al., 2009) runs in time  $\left(|\Omega|\right)^{O(1)}$  where  $\Omega$  is the universe. In our case,  $\Omega = \mathbb{V} \times \mathcal{V}$  hence the running time is  $O(C|\mathbb{V}|)^{O(1)}$ . The rest of the Algorithm 5 takes  $O(C|\mathbb{V}|)^{O(1)}$  steps, thus we obtain the stated running time and finish the proof.