I have a set of wireless links. These links are denoted by $\mathcal{L}=\{\ell_1, \dotsc, \ell_n\}$. Every link $\ell_i$ is composed of one transmitter $s_i$ and one receiver $r_i$.
Initially, all links are in the "sleep" mode (inactive). Whenever a link $\ell_i$ got activated, i.e., $s_i$ is sending a message to receiver $r_i$, this link will cause interference to all other active links (for now, $\ell_i$ will cause interference to $0$ links since no one else is activated).
Now, say, link $\ell_i$ is activated along with link $\ell_j$ for $i\neq j$. Then, $s_i$ will cause intereference to $r_j$ and $s_j$ will cause interference to $r_i$. This process continues until some number of links got activated, say, $\mathrm{N}$.
The problem is each receiver has a minimum allowed interference. i.e., receiver $r_i$ allow to receive an overall interference of $\kappa_i$. Let denote the interference at $r_i$ by $I_i$. Then the following condition must holds: $I_i\leq \kappa_i$. (note that $I_i$ is the summation of the interference generated by all other active links other than $i$).
This the problem: Activate as much link as possible while guaranteeing that all the receivers has the minimum allowed interference. Maybe like this: $$\max_{\ell} |\mathcal{L}|$$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;$subject to: $$I_i\leq \kappa_i$$
My question is: Why we cannot model this problem with graph theory? Why we cannot even model it with hypergaraphs? I also tried to model it using set theory but I also failed.
Note: I will present simple example why we cannot model it with graphs. Let say we have three links $\ell_1$, $\ell_2$, and $\ell_3$. If we model this by a complete graph of three vertices saying that each link causes interference to others, we will probably miss that two of these links can be activated together with no problem. One can activate $\ell_1$ with $\ell_2$, $\ell_2$ with $\ell_3$, and $\ell_1$ with $\ell_3$ but not all together.
Please any suggestions on how to model this problem mathematically?
EDIT1: Change the title according to Biderman's comment.
EDIT2: Write the problem according to Pepper's comment.
So, I can see one way to model this using a hypergraph. I have very little working knowledge in optimisation-type problems, so I don't know if this model will help: I'm hoping that someone else will give a better answer at some point.
We have a set of transmitter-receiver pairs, each pair endowed with coefficients $i_m$, $\kappa_m$, where $i_m$ is the interference produced by the transmitter, and $\kappa_m$ is the maximum interference allowed at the receiver.
We want to turn on as many transmitters as possible, but by turning on a transmitter we increase the interference, and potentially decrease the interference tolerance level (if the tolerance of the receiver we turn on is lower than the minimum of the receivers currently turned on).
Let $G$ by a hypergraph with one vertex for each transmitter-receiver pair. Define the hyperedge set using the following rule: $e$ is a hyperedge iff all of the vertices contained in $e$ can be active simultaneously (for any given set $A$ of vertices, this can be calculated quickly, by checking whether $\sum_A i_m \leq \min_A(\kappa_m)$).
Observe that this hypergraph has the property that every subset of an edge is an edge.
The problem is now equivalent to finding the largest edge in the hypergraph.
I'm not entirely sure that this "model" helps with the calculations however, because it really doesn't tell you anything that you didn't already know - you will still need to search for hyperedges in an exhaustive fashion (although you could use the above observation to reduce the work significantly).
One naive algorithm that springs to mind for solving it is as follows:
This algorithm is totally ignorant of the nature of the constraints determining whether a given hyperedge is present or not (namely how the linears sums and minimum function behave) - a more clever and efficient solution might arise from analysis of these things.
If in fact each transmitter produces a different amount of interference on each receiver, the above model and algorithm still works, but checking whether a given hyperedge is present will be slightly more complex.
I hope this helps.