It’s almost Christmas time and Santa is probably very busy with some last-minute preparations before his longer-than-7.5-million-kilometer trip around the world. One of the many things he has to worry about is how to pair up his reindeer in front of the sleigh. We all know that Rudolf goes right in front of everyone else because of his shiny nose, but what about his other eight four-legged friends? The traditional Christmas carols tell us that the reindeer are typically arranged in four pairs, front to back, as follows:
Therefore, we are going to assume that this is an arrangement that works pretty well (after all it’s been working since 1823). As someone with a degree in a STEM field (he wouldn’t reveal which, though), Santa can’t stop thinking about this interesting question: “Are there other good ways to pair up my reindeer?” Before we can answer that question, we need to define what a “good” pairing of reindeer is. After working tirelessly on Christmas eve, Santa’s reindeer have all the other 364 days of the year to hang out and get to know each other. As in every group of friends who spend a lot of time together, some friendships become closer than others. So it’s reasonable to expect that Rudolf’s eight friends will have a favorite companion for side-by-side galloping, a second favorite, a third favorite, etc. In addition, there’s one more important detail when it comes to reindeer pairings, according to Mrs. Claus: some of them like to be on the left side (Dasher, Prancer, Comet, and Donner), while others prefer to ride on the right side in front of Santa’s sleigh (Dancer, Vixen, Cupid, and Blitzen). Before you mention that I should also consider that male reindeer would rather be side-by-side with female reindeer, there’s scientific evidence that all of Santa’s reindeer are female, so we don’t have to worry about that.
After a nice conversation in front of his cozy fireplace, Santa was kind enough to provide me with the following lists of pairing preferences for each of his reindeer; though he vehemently asked me not to show any of this to his furry friends. I’m counting on you, my readers, to keep these lists to yourselves! The names in each list are sorted in decreasing order of pairing preference. The lefties appear in blue, while the righties appear in red (any resemblance to US political parties is a mere coincidence):
Dasher: Dancer, Cupid, Vixen, Blitzen
Prancer: Vixen, Blitzen, Dancer, Cupid
Comet: Cupid, Dancer, Blitzen, Vixen
Donner: Blitzen, Vixen, Dancer, Cupid
Dancer: Prancer, Comet, Dasher, Donner
Vixen: Dasher, Donner, Prancer, Comet
Cupid: Prancer, Dasher, Comet, Donner
Blitzen: Comet, Prancer, Donner, Dasher
Note that if we were to adhere to the lefties’ first picks, we’d end up with the traditional line-up. We are now ready to define what a good pairing is: a pairing is good (a.k.a. stable) if no one has an incentive to change pairs. In other words, if A is paired up with B, and A prefers C to B, it so happens that C, who is paired up with D, prefers D to A. (Note: this problem is known in the literature as the stable marriage problem and it arises in real life, for example, in the context of the National Resident Matching Program, which pairs up medical residents with hospitals every year in the United States.) Obviously, the traditional pairing shown above satisfies these goodness/stability conditions, given the reindeer’s preferences.
What Santa would like to know is whether or not there are other good pairings in addition to the traditional one. If so, he can add some variety to his line-up and the reindeer won’t get so bored by galloping side-by-side with the same companion every year. How can we help Santa answer this question? Using Operations Research, of course! More precisely, Constraint Programming (CP).
Constraint Programming is a modeling and solution paradigm for feasibility and optimization problems that allows one to represent complicated requirements (such as the stability condition above) in ways that are often easier and simpler than using traditional O.R. techniques such as Integer Programming. For example, indexing variables with variables and expressing logical constraints such as implications are a piece of cake in CP. Here’s a CP model written in the Comet language (not to be confused with Comet the reindeer) that answers Santa’s question. It essentially enforces the stability condition for every choice of A, B, C, and D.
The good news is that, in 3 milliseconds, that CP model finds all of the five different stable pairings. Here they are:
Update (1/1/2012): Here’s an AIMMS version of the CP model, kindly created and provided by Chris Kuip. Look for this reindeer example, including an accompanying graphical user interface, in an upcoming update to the set of examples in AIMMS.
I hope Santa reads this blog post before Christmas eve, but in case he doesn’t, please tell him to check this out if you run into him this holiday season. I’m sure his reindeer would appreciate a little change after 189 years.