A question on OR Stack Exchange asks how to number vertices in a layered graph so that the endpoints of edges have similar numbers. Note that "numbering" the vertices here means numbering *within layers* (so that, for instance, every layer has a vertex numbered 1). We will assume that every node has a unique ID before we launch into the numbering process. The author of the question chose the sum of squared differences of endpoint labels for all edges as the objective function to minimize. The following image shows an example of a layered graph with a (probably suboptimal) numbering scheme. The numbers on the edges are their contributions to the objective function.

The prolific Rob Pratt noted that the problem can be modeled as an assignment problem with a quadratic objective function, using binary variables. That model produces an exact solution, given sufficient time and memory.

Note that numbering the nodes within a layer is equivalent to picking one of the possible permutations of the node IDs. The author of the question indicated receptiveness to a metaheuristic, so I decided to try coding a random key genetic algorithm (RKGA) for the problem. I've mentioned RKGAs before (for instance, here and here). As I understand it, they were originally designed for sequencing / scheduling problems, where things need to be permuted optimally, so an RKGA seemed like a natural choice. I coded both the integer programming (IP) model and the RKGA in Java, using CPLEX 20.1 as the IP solver and Watchmaker Framework 0.7.1 for the GA. The Watchmaker Framework has not been under active development for quite a few years, but it works well.

To use an RKGA, you need to come up with a coding for a "chromosome" (candidate solution) and a mechanism for decoding the chromosome into a solution to the original problem (in this case separate vertex permutations for each graph layer) such that the decoded chromosome is always feasible. I chose as my chromosome representation a double-precision vector with elements between 0 and 1, having one element per vertex in the original graph. Double-precision is probably overkill, but I'm in the habit of using double-precision rather than single-precision, so it was the path of least resistance. To decode the chromosome, I first had to chop it up into smaller vectors (one per layer) and then extract the sort index of each smaller vector. So, using the image above as an example, a chromosome would be a double vector with 2 + 5 + 3 = 10 components. If the last three components were (0.623, 0.021, 0.444) the sort indices would be (3, 1, 2), yielding the vertex numbering for the last layer in the image. To convert a double vector into a sort index vector, I used a Java library (ValueOrderedMap, described in this post) that I wrote some time back.

A GA can "stagnate", meaning cease to improve on the best known solution. One obvious reason for stagnation is that it cannot improve on an optimal solution, but stagnation can occur for other reasons. On small test problems, the GA tended to stagnate rather quickly, so I set a stagnation limit and put the GA in a loop that would restart it up to nine times or until a specified time limit (five minutes ... I wasn't very ambitious). On larger test problems, it was not as quick to stagnate, but it eventually did.

I used the GA's best solution as a starting solution for the IP model. On smaller test problems, CPLEX occasionally closed the gap to zero within five minutes but usually did not. On larger problems, CPLEX struggled to get the best bound above zero (100% gap) within five minutes. So I suspect the IP model is prone to loose bounds. An example of a "small" test problem is one with 29 vertices spread across five layers and 35 edges. (I worked mainly with sparse examples, to keep the size of the IP model down.)

Interestingly, in *none* of the trials did CPLEX improve on the initial solution provided by the GA. So it might be that the GA was consistently finding an optimum, although I cannot prove that. (On the few smaller instances where CPLEX reached provable optimality, the optimal solution was indeed the GA solution.) Note that, while the GA solution appeared optimal, that does not mean that each GA run produced an optimum. On the 29 vertex example mentioned above, CPLEX found a provable optimum with objective value 111. The GA ran 10 times (using about 92 seconds total), and of the 10 runs two stagnated at 111, five at 112, and three at 114. So even if the GA is prone to finding optimal solutions, it may require multiple starts to do so.

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