Covering a graph with M cliques maximizing total edges weight

Question

I am working on a problem that involves distributing a set of N supplements across a predefined number of meals (M) in a way that maximizes the total number of positive interactions and minimizes negative interactions between the supplements. Each interaction between the supplements is known beforehand (positive, neutral or negative: see the image below).

I'm seeking advice on the mathematical formulation of this problem and potential algorithms or tools that could be used to solve it. Here's what I came up with:

• We can build a graph G from table of compatibility, with edges 1/0/-1 for positive, neutral and negative interactions correspondingly
• Then we can remove -1 edges completely
• Now we have modified graph G' with 0/1 edges only (see the image below)

• And we want to cover it with M cliques (bc clique in this graph will guarantee no 'bad' connections), finding a coverage with maximum number of +1 green edges

This method is not very good because:

• it doesn't consider using -1 red edges at all while building meals
• it has a very big time complexity (hours or days for 1 calculation on ~20 supplements)

I'm open for any suggestions. Maybe there are existing algorithms for solving this problem, or tools (like solvers)? Or is there a name for this problem I could search algorithms for?

Answer 1

One approach is to solve this with tools from operations research and combinatorial optimization, and specifically, to view this as an instance of ILP or SAT.

Let $x_{c,i}$ be a variable that will be true (i.e., 1) if supplement $i$ is assigned to meal $c$, or false (0) otherwise. Let $y_{c,i,j} = x_{c,i} \land x_{c,j}$. Let $r_{i,j}$ be the reward if supplements $i,j$ are used in the same meal ($+1$ if they interact positively, $0$ if there is no interaction, $-1$ if they interact negatively). Then your goal is to maximize $\sum r_{i,j} y_{c,i,j}$, subject to constraints that require $x$ to be a valid solution.

These constraints can be expressed in SAT or ILP pretty easily:

• Exactly one of $x_{c,i}$ is true, for each $i$. (In ILP: $\sum_c x_{c,i} = 1$. In SAT, this is a 1-out-of-M constraint.)

• $y_{c,i,j} = x_{c,i} \land x_{c,j}$. (In ILP: see Express boolean logic operations in zero-one integer linear programming (ILP). In SAT, this is $(x_{c,i} \lor \neg y_{c,i,j}) \land (x_{c,j} \lor \neg y_{c,i,j}) \land (\neg x_{c,i} \lor \neg x_{c,j} \lor x_{c,i,j})$.)

Then you can use a SAT solver that supports pseudo-boolean constraints (e.g., Z3) or an ILP solver (e.g., Cplex, Gurobi) to solve this system. If you use a SAT solver, you might want to use binary search over the target value of the objective function, using the SAT solver to search for a solution that achieves that target.

(Optionally, you can add some additional constraints for symmetry breaking, e.g., WLOG you can force $x_{1,1}$ to be true, force $x_{1,2} \lor x_{2,2}$ to be true, etc.)

If $N,M$ are big enough, this won't work, but it might be a simple to implement approach if $N,M$ aren't too big.