What is the relationship between "multi-camera" and laugh tracks?

It's a matter of logistics. Let's assume the producer wanted the finished episode to include the sounds of an audience laughing.

• A multi-camera sitcom could have a "live studio audience" to laugh at the jokes, because the scene is shot from multiple angles simultaneously and edited together later. The actors and camera operators go through the scene from beginning to end (usually), acting out the whole thing like a scene from a play. There may be multiple takes of the same scene, but the audience would still get a mostly-complete scene that would genuinely make them laugh. An artificial laugh track, added during post-production, would not be required.

• A single-camera sitcom would need to be shot multiple times from multiple different angles, with significant time between each angle, often with retakes. The actors and camera operators do not typically do the entire scene from beginning to end. This would ruin the humor for any "live studio audience", so the fake laughter of a laugh track would need to be added after shooting was done, during post-production.

Actually, probably all "laugh track" sitcoms, which are often shot before a "live, studio audience" use multiple cameras. (This is partly because they are shot in studios, which are designed to allow multiple cameras to move around during shooting.)

Non-laughtrack shows such as Curb Your Enthusiasm and Louie are sometimes termed "single camera shows" as they are set in real world locations and often use the techniques of "Cinéma vérité".

The article specifically mentions Modern Family which also uses a "single camera, documentary format", but likely utilizes multiple cameras per it's higher budget.

The article actually confused the term (or the OP). Having had a chance to talk to some producers I now understand: "Multi-camera" productions are those that are shot live, or in front of an audience. The idea is that you only get one take, and so you shoot it from all the angles you might want so that you can cut to different cameras. In a live broadcast cuts to different cameras are in fact done live. For shows with an audience one may in practice get multiple takes on a scene, and all the cuts are selected in post-production, but it's based on the same idea of getting a "live" performance.

On the other hand, "single-camera" productions are done for the camera. They are directed knowing that all shots are going to be edited in post into a final production that will only be watched on a single screen. So every shot is blocked and lit for the primary camera, and multiple takes of every shot are the norm. (Of course, large productions will bring multiple cameras onto a set, especially for large and expensive scenes. But the concept is still a production from a single camera.)