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I have weighted graph for sensor networks with aggregation nodes and sensors. There is a edge between two nodes associated with a weight. Higher the weight, stronger the interference between the aggregation nodes if they have same color.

I need to perform graph coloring of these nodes with just K colors. The nodes with higher edge weight should/must have different colors.

What is an efficient way to do the optimal coloring?

Ideas:

May be the coloring of such a big network is too difficult. Instead we can have clusters of nodes and then do the intra-cluster coloring. But then we need to take care of inter-cluster interference.

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I recommend first trying to solve the problem directly. For each edge $(i,j)$, let $c_{ij}$ be the edge weight. Let binary decision variable $x_{ik}$ indicate whether node $i$ is assigned color $k$. The problem is to minimize $$\sum_{i,j} c_{ij} \sum_k x_{ik} x_{jk}$$ subject to $\sum_k x_{ik} = 1$ for all $i$. You can call an MIQP solver or linearize the quadratic objective. To linearize, for each edge $(i,j)$ with $c_{ij}>0$, introduce binary (or just nonnegative) decision variable $y_{ij}$ to represent $\sum_k x_{ik} x_{jk}$, impose additional constraints $y_{ij} \ge x_{ik} + x_{jk} - 1$, and minimize $\sum_{i,j} c_{ij} y_{ij}$.

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  • $\begingroup$ you can see that there are large number of nodes. As it is clear from the problem description that one node will more more likely 5 to 6 nodes, i.e., most of $c_{i,j}$ are zero. How to take this into consideration o make the problem smaller/simpler? $\endgroup$
    – KGM
    Commented Apr 13, 2023 at 13:49
  • $\begingroup$ I would not be surprised if this problem can be solved quickly for a sparse graph with 500 nodes. You do not need $y_{ij}$ or its constraint if $c_{ij}=0$. I'll update my answer. $\endgroup$
    – RobPratt
    Commented Apr 13, 2023 at 13:54
  • $\begingroup$ This is the same model I suggested here: or.stackexchange.com/a/10238/500 $\endgroup$
    – RobPratt
    Commented Apr 13, 2023 at 14:43
  • $\begingroup$ the second constraint can be expressed as: for $c=1:C$, for $i=1:N-1$, for $j=i+1:N$, if $w_{ij}>0$ $y_{ij}\ge x_{ic}+x_{jc}-1$, end if, end for, end for, end for. Am I right? $\endgroup$
    – KGM
    Commented Apr 26, 2023 at 10:51
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    $\begingroup$ Yes, this model allows such improper colorings. For edge $(i,j)$, if nodes $i$ and $j$ are assigned the same color $k$, then $x_{ik}=x_{jk}=1$, the constraints enforce $y_{ij}=1$, and the weight $c_{ij}$ is accrued. $\endgroup$
    – RobPratt
    Commented May 11, 2023 at 12:56

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