Hello,

I am in the process of building some software to generate fixtures for a chess league. Which has a little twist which complicates matters. I would like to introduce a constraint. Where by a team and the below team from a club are not allowed to play on the same night. e.g. Loughborough 1 is not allowed to play on the same night as Loughborough 2. This help teams share players, therefore play more games.

The league has five divisions, with 8, 7, 8, 7, 7 teams in respectively. The first half of the league is played over 10 weeks. This means that teams do not have to play each week.

A solution would be to brute force all the fixture combinations, for all the divisions. The problem with this approach is that there are way too many combinations!

I am wondering if there are any mathematical techniques that I could use to help me with this problem. I am not wanting to find a unique solution (a combination of fixtures, without violating the constraints). I would be happy with an algorithm which produced 10 violation of constraints in the season.

Any help or pointers would be greatly appreciated.

Can you suggest any areas for me to research.

Thanks in advance.