I'm trying to find An efficient algorithm for the following problem:

I have $n$ sets of numbers (different sets might have common numbers) and I need to find union combination of $k$ sets that Will yield the biggest set.

For example: if i have the sets $A1=\{1,3,4\}, A2=\{1,2,4\}, A3=\{5,7\}$ in that case $n=3$ and lets say that $k=2$. $|A1 \cup A3| =5$ or $|A2 \cup A3| =5$ will yield the biggest set In contrast to $|A1 \cup A2|$ = 4.

If this is np hard question (i don't know if it is) a good Approximation will be fine I think that brute force is $O({n \choose k})$ operations.

  • $\begingroup$ This is maximum coverage problem, an NP-hard problem. A greedy algorithm that chooses a set which contains the largest number of uncovered elements is $0.63$-approximation. $\endgroup$ – zdm Nov 7 '19 at 17:14
  • $\begingroup$ This looks very much like the Set Cover problem when you change your problem if you can cover all the numbers with k sets. For this problem there is no general constant factor approximation but one using the frequency as approximation factor ( in how many sets does a number appear?). I don't know how it generalizes to your problem where you want to maximize the covered numbers. $\endgroup$ – Daniel Nov 7 '19 at 17:20
  • $\begingroup$ we are speaking about approximately 10000 sets $\endgroup$ – Barak Mordehai Nov 7 '19 at 18:40

Your problem is known as the Maximum coverage problem. If $S$ is a solution (a set of at most $k$ sets), then an element is covered if it belongs to $\cup_{X \in S} X$. You want to find a solution that maximizes the number of covered elements. This problem is known to be $\textrm{NP}$-hard (see below for a possible reduction) and admits a greedy approximation algorithm that will cover at least a $\left( 1- \frac{1}{e} \right)$-fraction of the elements covered by an optimal solution.

In addition, there is an easy reduction from $3$-SAT to your problem.

Consider an instance of $3$-SAT with $n$ variables $x_1, \dots, x_n$ and $m$ clauses $C_1, \dots, C_m$. You can obtain an instance of your problem by creating two sets $T_i$ and $F_i$ for each variable $x_i$. The sets are defined as follows:

  • $T_i = \{ j \mid x_i \in C_j \} \cup A_i$ ;
  • $F_i = \{ j \mid \overline{x_i} \in C_j \} \cup A_i$,

where $A_1, \dots, A_n$ are arbitrary pairwise-disjoint sets, each containing $m+1$ integers between $m+1$ and $m+n(m+1)$.

Deciding whether you can select $n$ sets such that their union has cardinality at least $m+n(m+1)$ is now equivalent to deciding whether the $3$-SAT formula is satisfiable.

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  • $\begingroup$ first of all thank you for the answer (1−1/e) -fraction is pretty low, is there average fraction i just want to know if i can count on it because its connected to my work $\endgroup$ – Barak Mordehai Nov 7 '19 at 18:29
  • $\begingroup$ What do you mean by "is there average fraction"? I am not an expert on this particular problem so I don't know more than what can be found online... The wikipedia page I linked claims that "lnapproximability results show that the greedy algorithm is essentially the best-possible polynomial time approximation algorithm for maximum coverage unless P = NP" and cites this paper by Uriel Feige. $\endgroup$ – Steven Nov 7 '19 at 19:09

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