If you do a simple google search or look through one of several lists of swiss tournament scheduling programs out there, you'll find that there is no shortage in this arena due to the abundance of chess, go, and card tournaments using this format. Swiss scheduling is clearly a difficult problem and is one that a lot of people attempt to build programs for. My first instinct was thus to use an existing program and just manually add a bunch of games to Leaguevine each round during the tournament. Yeah, I know, keeping track of games on both Leaguevine and an external scheduler sounds awful. What makes this approach even worse is that after looking into these existing programs, they tended to do a very poor job of scheduling because they did not ensure that there were no rematches.
Thus, we built our own.

Before I explain our solution, I should briefly explain how a swiss tournament works. Each round, all of the teams are matched up with another team based on a pairing method that the tournament director chooses. The winners of that round received points which are used to rank the teams. Typically, a team is rewarded one point (we'll refer to this as a "swiss point") for a win, but there also exist variations based on point differential. If multiple teams have the same number of swiss points, the tiebreaker is usually how many swiss points each team's opponents have scored, which essentially rewards teams for their strength of schedule.

To schedule a round of matchups, the teams are first separated into groups based on either their win/loss record or swiss points. Unless point differential is taken into account, the groups will likely be the same for win/loss record and for swiss points. If point differential is taken into account, then the groups are formed based off of win/loss record and the point differential (aka swiss points) determines the team's ranking within that group. The teams will then play the other teams in this group. If a group has an odd number of teams, a team must be added or removed from that group. For deciding which team plays which other team within the group, there are four common group pairing methods.

The difficulty in scheduling occurs when we introduce two almost universally accepted constraints:

1. No team should play another team more than once during the tournament
2. No team should have more than one bye during the tournament

Because of this, the matchups that are generated by a tournament scheduler using any of the group pairing techniques will result in matchups that violate either of these two constraints. Almost every piece of existing software I've seen simply points out which ones violate the constraints, but leave it entirely up to the tournament director to resolve the problems. There does appear to be one program   out there that resolves these conflicts, but I didn't notice it until well after I finished coding the algorithm up. Luckily it only took a couple days!

I've seen many articles explaining how to resolve any conflicts with these constraints. They tend to list a sequence of steps for either changing the matchups until you find something that works or reordering the standings until you find something that works. However, at the end of these steps usually comes a line that says "this approach usually works but doesn't work all of the time.". Another problem I see with these approaches is that it is very difficult for a person to manually change a matchup and have that change be the optimal possible change. There is just too much room for error.