Short summary: The problem is to assign people to meetings at different days while respecting the capacities and (inter-day) constraints of the meetings. Each person can only attend at most one meeting at a day.
I don't know if this problem has a common name - therefore the weird title question, sorry. If I overcomplicated any part of the problem definition, let me know.
People P = $\{p_1, .., p_n\}$, Meeting days D = $\{d_1, .., d_m\}$,
Meetings for each day: $M_{d_1} = \{m_{d_1,1}, .., m_{d_1,a}\}, M_{d_2} = \{m_{d_2,1}, .., m_{d_2,b}\}, .., M_{d_m} = \{m_{d_m,1}, .., m_{d_m,z}\}$
The $M_{d_i}$'s are pairwise disjoint.
Each meeting $m$ can hold a maximum number of people of $\text{cap}_{m}$ (capacity),
A person must not attend two (or more) meetings at different days that are "similar". (These conflicts are only between different days ("inter-day conflicts"). There are no conflicts within a day.)
Therefore define meeting constraints $C=\{c_1,..c_q\}$. Every meeting $m$ has exactly one corresponding constraint $c_i = \text{const}(m)$, s.t. [$\text{const}(m_i) = \text{const}(m_j)$] iff. [$m_i$ is "similar" to $m_j$] for all meetings.
There are preferences which person wants to attend which meeting at a particular day, e.g.:
(here 2 preferences are specified - if we generalize it to a fixed $K$ does the problem get any harder with regards to NP-hardness etc?)
$\text{Prefs} = \{$
$(p_1, \color{red}{d_1}, m_{\color{red}{d_1}, \color{green}{2}}, m_{\color{red}{d_1}, \color{green}{4}}),$ // Person 1 wants to attend the $\color{green}{2nd}$ or $\color{green}{4th}$ meeting at $\color{red}{day 1}$
$(p_1, d_3, m_{d_3, 1}, m_{d_3, 2}),$
$(p_3, d_2, m_{d_2, 2}, m_{d_2, 3}),$
$..$
$\}$, where:
a) every person-day combination $(p_i, d_j, ..)$ appears at most once.
b) every tuple has the form: $(p_i, \color{red}{d_j}, m_{\color{red}{d_j}, a}, m_{\color{red}{d_j}, b})$, with $a \neq b$ ($\widehat{=}$ valid preferences only)
(Note: The meeting preferences for each person-day combination are not supposed to be in any order, so (.., .., $x$, $y$) does not mean $x$ is preferred over $y$.)
A valid solution to the problem is:
Assign each person-day combination $(p_i, d_j, X, Y) \in \text{Prefs} $ to the meeting X or the meeting Y or no meeting at all, s.t.:
a) for every meeting $m$: we don't assign more people to $m$ than its capacity.
b) for every person $p$: for every constraint $c$: the person $p$ is assigned to (at most) one meeting with constraint $c$. (Otherwise, $p$ would attend two meetings at different days that are "similar".)
We want to maximize the number of person-day combinations that are assigned to a meeting. (So minimize the number of person-day combinations that are assigned to no meeting at all.)
Is this problem in $P$? (If yes, what is the idea for a polynomial-time algorithm?)
Is this problem NP-hard? (I tried a reduction from Bin-Packing $\leq_p$ ThisProblem, but without success. I think it's NP-hard and would appreciate any helpful ideas for reductions.)