round the element $e_i^{*}$ defined as $e_i^{*} =  \operatorname*{arg\,max}_{ e\in V}  f_i({S_{\rm{tr}}}\cup S_i\cup\{e\})-f_i({S_{\rm{tr}}}\cup S_i)$ is added to $S_i$. In the second phase, the set ${S_{\rm{tr}}}$ is formed greedily in $l$ rounds, and in each round the element $e^{*}  $ defined as $e^{*} =  \operatorname*{arg\,max}_{ e\in V} \sum_{i=1}^{m}f_i({S_{\rm{tr}}}\cup\{e\}\cup S_i)-f_i({S_{\rm{tr}}}\cup S_i)$ is added.

**Initialize** the sets ${S_{\rm{tr}}}$ and $\{S_i\}_{i=1}^m$ to the empty set. Find $e_i^{*} =  \operatorname*{arg\,max}_{ e\in V}  f_i(S_i\cup\{e\})-f_i(S_i)$ $S_i\xleftarrow{} S_i\cup\{e_i^{*}\}$ **end for** **end for** Find $e^{*} =  \operatorname*{arg\,max}_{ e\in V} \sum_{i=1}^{m}f_i({S_{\rm{tr}}}\cup\{e\}\cup S_i)-f_i({S_{\rm{tr}}}\cup S_i)$ ${S_{\rm{tr}}}\xleftarrow{} {S_{\rm{tr}}}\cup\{e^{*}\}$ **end for**

return ${S_{\rm{tr}}}$ and $S_i$

While the solutions obtained by Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] are guaranteed to be near-optimal, it turns out that they can be complementary with respect to each other. Our *main* deterministic algorithm, called `Meta-Greedy`, runs both Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] and chooses as output the solution, among the two, that leads to a higher objective value in . To be more specific, if we consider ${S_{\rm{tr}}}^{(1)}, \{S_i^{(1)}\}_{i=1}^m$ as the outputs of Algorithm \[alg: discrete meta-Greedy\] and ${S_{\rm{tr}}}^{(2)}, \{S_i^{(2)}\}_{i=1}^m$ as the outputs of Algorithm \[alg: Reverse discrete meta-Greedy\], then `Meta-Greedy` compares the values of $ \sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{(1)}\cup S_i^{(1)})$ and $ \sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{(2)}\cup S_i^{(2)})$ 