Suppose I have some set of events $E$, each associated with a collection of time periods $P$ that these events occur, where $P_1$ is earlier than $P_2$. For example:
$$ \begin{align} E_1 &\rightarrow \{P_1, P_2, P_3\} \\ E_2 &\rightarrow \{P_4, P_8\} \\ E_3 &\rightarrow \{P_3\} \end{align} $$
I want to schedule each event once, while also maximizing the time between these events. So in this case, is there an algorithm to find the matches:
$$ \begin{align} P_1 &\rightarrow E_1 \\ P_3 &\rightarrow E_3 \\ P_8 &\rightarrow E_2 \end{align} $$
This is optimal because it maximizes the time between events ($2 + 5$), essentially spreading out these events as much as possible. Note that the ordering $P_1, P_4, P_8$ is more desirable than $P_1, P_2, P_8$ because it divides space evenly even though the time between the first and last periods is the same.
The matches that minimize the time between events can be found with bipartite graphs, but can this be generalized to maximization?