GIVEN: Positive integers $n,m,L$ and probabilities $p_1, p_2, \ldots, p_n$.

GOAL: Choose $L$ size-$m$ subsets $S_1, S_2, \ldots, S_L$ of $\{1,2,\ldots,n\}$ to maximize $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ]$, where the expectation is taken over (all $2^n$) $S \subseteq \{1,2,\ldots,n\}$ using the pdf $f$ defined as \[ f(S) = \left( \prod_{i \in S} p_i \right) \left( \prod_{i \not\in S} (1-p_i) \right)\ . \]

(Equivalently, $S$ is chosen by examining each $i \in \{1,2,\ldots,n\}$ independently, choosing $i$ to be in $S$ with probability $p_i$.)

Note: the problem is trivial unless $m < n$ and $L < \binom{n}{m}$.

A nice, closed-form expression for $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ]$ (in terms of $n,m,L$ and the $p_i$ and the $S_\ell$) would be a good start.

EDIT:

WLOG, $p_1 \geq p_2 \geq \cdots \geq p_n$.

Special cases:

$L=1$: $S_1 = \{ 1, 2, \ldots, m \}$ maximizes $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ]$.

$m=1$: Denote by $x_i$ the single element of $S_i$ (i.e., $S_i = \{ x_i \}$) and $X = \{ x_1, x_2, \ldots, x_L \} = \bigcup S_i$. $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ] = \sum_{S \cap X \neq \emptyset} 1 \cdot f(S) = 1 - \sum_{S \subseteq \{ 1, 2, \ldots, n \} \setminus X} f(S) = 1 - \prod_{i \in X} (1-p_i)$, which is maximized by the greedy solution of $S_i = \{ i \}$ (for $1 \leq i \leq L$).

Here's how I examined a few more special cases:

This example ($(n,m,L)=(4,1,2)$) falls under the already discussed $m=1$ case, but I present it to hopefully make the presentation of subsequent examples clearer:

$\binom{\binom{n}{m}}{L} = \binom{\binom{4}{1}}{2} = 6$ possibilities for $(S_1,S_2)$:

```
1100 1ooo,o1oo, p1+p2 - p1p2
1010 1ooo,oo1o, p1+ p3 - p1p3
1001 1ooo,ooo1, p1+ p4 - p1p4
0110 o1oo,oo1o, p2+p3 - p2p3
0101 o1oo,ooo1, p2+ p4 - p2p4
0011 oo1o,ooo1, p3+p4 - p3p4
```

- Column 1: the sum of the subsequent $L=2$ columns (I used "o" for "0" in those for readability)
- Column 2: represents $S_1$ (the 1st bit is 1 iff $1 \in S$ (0 otherwise), the 2nd bit is 1 iff $2 \in S$, etc.)
- Column 3: represents $S_2$
- Column 4: $\displaystyle \mathbb{E}[ \max_{1\leq \ell \leq L} |S \cap S_\ell| ]$ (formatted to put the degree-1 terms all together first, then the degree-2 terms, etc., with spacing so you can look up and down and see which other rows share those terms).

$\binom{\binom{n}{m}}{L} = \binom{\binom{4}{2}}{2} = 15$ possibilities for $(S_1,S_2)$:

```
2110 11oo,1o1o, p1+p2+p3 - p2p3
2101 11oo,1oo1, p1+p2+ p4 - p2p4
1210 11oo,o11o, p1+p2+p3 - p1p3
1201 11oo,o1o1, p1+p2+ p4 - p1p4
2011 1o1o,1oo1, p1+ p3+p4 - p3p4
1120 1o1o,o11o, p1+p2+p3 - p1p2
1021 1o1o,oo11, p1+ p3+p4 - p1p4
1102 1oo1,o1o1, p1+p2+ p4 - p1p2
1012 1oo1,oo11, p1+ p3+p4 - p1p3
0211 011o,o1o1, p2+p3+p4 - p3p4
0121 011o,oo11, p2+p3+p4 - p2p4
0112 01o1,oo11, p2+p3+p4 - p2p3
1111 11oo,oo11, p1+p2+p3+p4 - p1p3-p1p4-p2p3-p2p4 + p1p2p3+p1p2p4+p1p3p4+p2p3p4 - 2p1p2p3p4
1111 1o1o,o1o1, p1+p2+p3+p4 - p1p2- p1p4-p2p3- p3p4 + p1p2p3+p1p2p4+p1p3p4+p2p3p4 - 2p1p2p3p4
1111 1oo1,o11o, p1+p2+p3+p4 - p1p2-p1p3- p2p4-p3p4 + p1p2p3+p1p2p4+p1p3p4+p2p3p4 - 2p1p2p3p4
```

I separated the rows into two groups; two rows are in the same group if their first-column entries are permutations of each other.

Notes: the coefficient of 2 in the degree-4 terms of the last three rows seems interesting to me. The sign of a term is the negation of $(-1)$ to the degree of the term.

$\binom{\binom{n}{m}}{L} = \binom{\binom{4}{2}}{3} = 20$ possibilities for $(S_1,S_2,S_3)$:

```
3111 11oo,1o1o,1oo1, p1+p2+p3+p4 - p2p3-p2p4-p3p4 + p2p3p4
1311 11oo,o11o,o1o1, p1+p2+p3+p4 - p1p3-p1p4- p3p4 + p1p3p4
1131 1o1o,o11o,oo11, p1+p2+p3+p4 - p1p2- p1p4- p2p4 + p1p2p4
1113 1oo1,o1o1,oo11, p1+p2+p3+p4 - p1p2-p1p3- p2p3 + p1p2p3
2220 11oo,1o1o,o11o, p1+p2+p3 - p1p2p3
2202 11oo,1oo1,o1o1, p1+p2+ p4 - p1p2p4
2022 1o1o,1oo1,oo11, p1+ p3+p4 - p1p3p4
0222 o11o,o1o1,oo11, p2+p3+p4 - p2p3p4
2211 11oo,1o1o,o1o1, p1+p2+p3+p4 - p1p4-p2p3- p3p4 + p1p3p4+p2p3p4 - p1p2p3p4
2121 11oo,1o1o,oo11, p1+p2+p3+p4 - p1p4-p2p3-p2p4 + p1p2p4+ p2p3p4 - p1p2p3p4
2211 11oo,1oo1,o11o, p1+p2+p3+p4 - p1p3- p2p4-p3p4 + p1p3p4+p2p3p4 - p1p2p3p4
2112 11oo,1oo1,oo11, p1+p2+p3+p4 - p1p3- p2p3–p2p4 + p1p2p3+ p2p3p4 - p1p2p3p4
1221 11oo,o11o,oo11, p1+p2+p3+p4 - p1p3-p1p4- p2p4 + p1p2p4+p1p3p4 - p1p2p3p4
1212 11oo,o1o1,oo11, p1+p2+p3+p4 - p1p3–p1p4-p2p3 + p1p2p3+ p1p3p4 - p1p2p3p4
2121 1o1o,1oo1,o11o, p1+p2+p3+p4 - p1p2- p2p4–p3p4 + p1p2p4+ p2p3p4 - p1p2p3p4
2112 1o1o,1oo1,o1o1, p1+p2+p3+p4 - p1p2- p2p3- p3p4 + p1p2p3+ p2p3p4 - p1p2p3p4
1221 1o1o,o11o,o1o1, p1+p2+p3+p4 - p1p2- p1p4- p3p4 + p1p2p4+p1p3p4 - p1p2p3p4
1122 1o1o,o1o1,oo11, p1+p2+p3+p4 - p1p2- p1p4-p2p3 + p1p2p3+p1p2p4 - p1p2p3p4
1212 1oo1,o11o,o1o1, p1+p2+p3+p4 - p1p2-p1p3- p3p4 + p1p2p3+ p1p3p4 - p1p2p3p4
1122 1oo1,o11o,oo11, p1+p2+p3+p4 - p1p2-p1p3- p2p4 + p1p2p3+p1p2p4 - p1p2p3p4
```

Note: the sign of a term is the negation of $(-1)$ to the parity of the degree of the term only in the first and third block of rows — the degree-3 terms in the 4 rows in the middle block are all negative.

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