# Relationship Formation as an Optimization Problem (Part 1?)

It’s been as while since I’ve updated and I figured why not to add a new post? And with that hamfisted introduction complete, I present the following: Buddy Pairing as a Constraint Satisfaction Problem (CSP).

## Background

Before I get too carried away at explaining the implementation, let me provide provide some context. The reason why I am even thinking about human relationships and constraint optimization in the same thought is that it’s tied to a work project. The project has a very simple goal: Automate the pairing of new hires and current full time associates together. Now the purposes of having this pairing exists is to support our new hires by assigning (which the vast majority are new graduates as well) someone that they can ask questions carte blanche, anything from how do I use the photocopier (if that actually applied right now…) to what should I think about for my long term career?

Now, as you probably guessed, any random pairing of new hires to associates would not do. Ideally, the pairings would be done so that these the buddies share as many of the same interests and preferences as possible so that both get the maximal benefit out of the relationship as possible.

Upon learning about the project and hearing that the project needed volunteers, I almost immediately nerd-sniped myself in wondering how would you go about doing that (or at least, in a somewhat elegant manner). And so here we are.

## Intermission for Some Definitions

Before I dive at the heart of the solution I landed on, it probably would be in my best interest to informally cover what a CSP is. If my very limited search of Reddit and Hacker News is anything to go by, constraint programming and CSPs are not exactly a popular topic, and it probably would be foolish to just assume everyone remembers or knows what a CSP is. Or at least, remembers enough beyond whatever they needed to pass a Leetcode or HackerRank interview.

So very quickly, Constraint Satisfaction Problem (CSP) is defined simply as a relation of three elements: variables, domains, and constraints. Now between our variables, the values in the domains allowable for our variables, and the constraints governing the variables and domains, there’s hopefully some kind of assignment or configuration of values in said variables that does not violate the constraints.

Now, in many scenarios, your constraints must always be satisfied and hold true, e.g. like graph coloring, Sudoku, equation solving, and so forth. However, for the use case I am working on, it would be a little inconvenient and impractical to have for a pairing to satisfy all constraints, especially if that constraint is something as diverse and varied as people’s preferences and interests. After all, what is the likelihood that I was going to always create pairings where said people shared everything in common?

Ideally, the constraint where the new hire and associate share common interests and preferences would have to be relaxed, made “soft” if you will. However, I can not just completely relax the constraint because otherwise we could skip this who process and do random assignments. So what if for each possible pairing, we assigned a cost to how much they do not share? So now the model has become a little more interesting in that it’s now a weighted CSP (WCSP). All that really just means is that now an assignment has an overall cost and as part of our solution we would like to either minimize or maximize that cost.

## The Semantics

Before we can model anything with a CSP, we need to actually define the problem we are trying to solve :-). For my particular use case, the requirements were pretty straightforward, and hence probably why I was actually come up with something that worked in a relatively straightforward fashion. The constraints are as follows:

• Pairs have to belong to the same location
• Pairs have to consist of a new hire and associate
• An associate may have multiple new hires assigned to them
• A new hire will only have one associate
• Pairs have to should share as many as the same interests and responses as possible

The constraints I actually bother as part of the model/CSP, only the last four wound up being explicitly defined as part of the model. The first constraint was just handled as part of the database query that was used to fetch all associates and new hires. It kept things nice and simple, and did not require me to think especially hard on how to add that constraint to the model. Plus gives me a nice easy bonus of being able to parallelize the solving operations over locations.

## The Tool

With the problem defined and a direction identified, it was now time to turn theory into code. To start, I needed to search for an optimization package or tool that will hopefully let me model this problem without needing me to implement the algorithms myself.

Now, one interesting wrinkle for the search was that the tool had be in a language that I was familiar with and that the language is also used at the company extensively. Also, the language ideally would also be one that the team also had plenty of experience with, and be the same one that we would build out other components in.

And thus began the quixotic search for an optimization package with a JavaScript interface. Unsurprisingly, I didn’t find much. As it turns out, Javascript is not exactly a popular language for doing constraint optimization with. I did briefly contemplate on using Node Native API so I could broaden my search to include tools implemented in C++ but that would have pretty much guaranteed any solution created would have been immediately been unmaintainable.

Fortunately, while contemplating in whether or not to learn how to use the Node Native API worked and how hard it be to learn how to code with V8, I found another a tool called OR-Tools. Maintained by some lovely folk over at Google, while the core of the package is still implemented in C++, they also provide a Python interface that’s generated with SWIG from the C++ source. So not quite what I was looking for, but close enough for my team and I to be happy.

## And Now, for Something that Did Not Work

Before I just go off and talk about what I wound up implementing, I want to briefly cover an attempt that did not work. As a matter of opinion, I feel it’s important to cover the path to get here, and show how the solution was not just preordained.

So the initial implementation I had of modeling the problem did not involve CSPs and minimizing an objective function, but rather a more naive idea where all the possible answers and responses were modeled as nodes in a graph, and then for both new hires and associates would have arcs connected between these answer nodes and that I would try to form a pairing by maximally matching these nodes together.

However, some brief research suggested this kind of tripartite matching would be computationally expensive and time consuming. However, this line of research did lead down to bipartite matching and learning about how bipartite matching reduces down to network flow problems. Upon seeing that, I was ecstatic, seeing as network flow is well covered and there are more than a few libraries out there that had network flow solver, among which OR-Tools counted as one. The plan was now to instead to consider a single new hire at a time and find a maximal matching between the new hire responses and all the full time associates and utilize the flow computed as the means of picking the best buddy. Sure, considering a single new hire at a time would probably produce solutions that were not global optimal, but hey, I just needed an assignment, not a perfect assignment. Whatever was created would be substantially better that the current process which was humans going through by hand to create buddy pairs.

Alas, and perhaps thankfully, the idea did not work out; I modeled the solver as described, ran it, and found that it reported a flow of zero and therefore did not produce anything useful. Now, I will admit that I did not dissect the reason as to why seeing as the project was now several weeks in and I needed to get this out the door, but it boiled down to the following: I made a number of bad assumptions. In my haste in seeing that the problem could be reduced to a network flow, I had missed a number of key conditions that allowed network flow be reducible to an assignment problem.

The core condition I believe is that I did not maintain is where flow heading into the nodes must the same amount headed out. From doing what I believe the graph should look like on paper, I saw that I had answer nodes with flow in but nothing out. This happens as sometimes none of the associates would share some of the answers that a new hire inputed. Now, on further reflection, the solution I believe would have been to add a dummy node that would have maintained that condition, but I am not entirely sure if would have worked great in any case. If nothing more, a good lesson in not being too hasty and making sure I have all the details.

However, now with this approach coming to screeching halt, it was time to think very hard and come up with a new approach.

## The New Approach

Now stuck with a potential model that did not work, and faced with the prospect that I might have over-promised a little bit, something occurred to me. I do not care about the specific answers that each buddy pair share, I just care about to what degree they do not share the same interests with each other. So instead of explicitly matching answers from a new hire to associates, what if I instead compared the answers that a new hire and an associates had, and determine what interests that the associate does not share with the new hire? In other words, the set difference between the new hire’s answers and the associate’s answers?

Cue a few days of obsessive tinkering and eureka! I now have a WCSP that generates useful pairings for me! So the problem/model semantically became the following:

• The variables are to be all the new hires, each new hire being considered a variable
• The domains for each variable/new hire are all the associates that share even a single answer with said new hire.
• For each associate/value in the domain for a specific new hire, the cost for that assignment to that each new hire/variable is the weighted sum of all the answers the associate does not have in common with the new hire
• The objective function we minimize over is simply the sum of the all the costs for each assignment of associates to new hires

### Quick Note on the Objective Function

Now this is probably not necessary to explain, but the objective function is defined as a weighted sum because in the future it would allow us to penalize possible pairings differently for not having certain answers versus others. So for example, if we did not want people who like pineapple on pizza to be paired with people who do not like pineapple on pizza, we could strongly penalizing that assignment by changing the associated cost with that answer or question.

Beyond the general idea, there’s a couple implementation details I want to mention. One small detail is that the cost I assigned to my problems are all of type integer. It’s partially for simplicity (like, what would a weight of 1.5 vs 2 get you?), and also because for many of the solvers in OR-Tools, it’s a requirement that your values are either representable as a boolean or integer value. If you find yourself wanting floating point values, do what this nice StackOverflow Answer says to do and just scale your values. Bonus is you do not have to worry about the effects of floating point precision in evaluating an optimal solution.

Another note about the objective function is that the choice to minimize versus maximize was an arbitrary choice and influenced by convention. You can easily define the objective function to use the set intersection of the new hire and associates answers and maximize the “cost”. I just choose to minimize my objective function because a lot of literature and examples just often talk about minimizing the objective/cost/loss function

### Quick Aside about OR-Tools Performance

With a working implementation in hand, one big question I did have to address was the performance of model, specifically how long it takes to solve as the number of new hires and associates increase? After all, I do have to deploy this to production, and end-users should not have to wait hours to get a result. Initial testing with mock data of ten new hires and seventeen associates did not yield promising news; in fact, even if I gave the model an hour to solve, it only found feasible solutions, not optimal ones. Sure, the model did create an assignment that pairs everyone, and for our purposes probably would’ve been okay since this was a completely manual process before this, but it still was disappointing to see.

Fortunately, I found out, in a circuitous manner, that the CP-SAT solver supported multi-threading. As part of the solver setup, you set a parameter called `solver.sat.num_search_workers` to tell it how many workers you want (eight was the number often suggested in threads on the OR-Tools Forums ) and boom, it made the solver go from hours and only feasible solutions to tenths of seconds and optimal solutions. Probably will be one of the few times in my career where the phrase “let’s use multithreading!” and not have it immediately complicate matters. Helps that I did not do any of the hard work, the fine people over at OR-Tools did. Granted, the production data would be larger still, but not significantly so. Will be interesting to see how well it all scales out.

Now, there is no such thing as a free lunch and as such, there are limitations. One, the solver does become non-deterministic. As highlighted in the release notes, it uses randomization to set different parameters and is something to be aware of. Another thing to be aware of is that the multi-threaded solver, as of time of writing, does not support finding of all solutions. For my application, these two limitations did not concern me since I only need a solution, but your mileage may vary.

## Conclusion

All in all, I’ve really enjoyed my time working on this project. Granted, this problem is not particularly hard, as evidenced by the fact I did not need to spend weeks banging my head against the wall trying to make the problem workable. On the bright side, I do get to say that I actually used what I am learned in degree in my job, a mildly interesting and also completely unimportant distinction to have.

Now, if you want to try this yourself, the assignment problem example and the nurse scheduling example I found exceptionally useful. The final code output, while not shown here, is extremely similar to the examples; the assignment example being useful in learning how to define a model and the nurse’s example was useful in demonstrating a more involved setup. The only real differences were a lot of utility functions to dynamically compute the cost (really, figuring out how to create the correct database MongoDB queries); that and a bit of work to create a multi-key/bidirectional data structure that let me use said data structure both like a list and dictionary at the same time.

As for the outcome of the project and it’s future, I’ve already gotten a few chances to use it for real at the expected dataset sizes and the tool has performed admirably, and the team has gotten useful feedback on the pairing process and the overall product which we are looking to incorporate. As such, the work continues on the buddy pairing process and it’s CSP.