Planning Tournament with p People & g Games: Algorithm Needed

[cubemx]

Hi,
I have a practical math problem that I can't solve by myself.
The problem is probably already solved by other people but I can't find it using Google.
If somebody knows a solution or is willing to try and solve it that would be awesome.
I need a algorithm to plan some kind of tournament.

This is the problem:
There are a total of p people, they are distributed over teams.
Ever round g different games are played, t teams play game A, t teams play game B etc. for all g games. (So the total number of teams is t*g.)

For instance:
g = 5, t=3 (so there a 15 teams)

....Round 1......Round 2.... Round 3 etc. etc.
Game A:...1 vs 2 vs 3….....4 vs 8 vs 15...…
Game B:...4 vs 5 vs 6...1 vs 7 vs 12...…
Game C:...7 vs 8 vs 9...2 vs 10 vs 13...…
Game D:...10 vs 11 vs 12...3 vs 6 vs 14...…
Game E:...13 vs 14 vs 15...5 vs 9 vs 11...…


Rule 1: Every team has to play every game at least once, but no more than strictly necessary to make all rules work.

Rule 2: Every team has to play against every other team,so team 1 has to play team 2,3,4...15 (team 2 has to play team all other teams etc. etc.). If team 1 already played team 12, they can only play each other again if it's strictly necessary to make the rules work. In the example above there was a match "1 vs 7 vs 12" so later on match "1 vs 12 vs 9" may only happen is there is no other option.

Rule 3: If a team has to play a game more than once, it's better if there is a pause between those games (so not in successive rounds).

Rule 4: If a team has to play a other team more than once, it's better if there is a pause between those games (so not in successive rounds).

Rule 5: If the people can't be distributed equally over the teams (for instance p=147 and there are 15 teams, three teams will be smaller), smaller teams shouldn't play each other if possible. So if team 13, 14 and 15 are smaller match "13 vs 14 vs 15" should be avoided if possible. (While match "13 vs 14 vs …" can't be avoided since team 13 has to play team 14 at least once.)

The rules are ordered according to importance, rule 1 is the most important.

The answer to this problem may be given in any form: group theory, graphs, algebra ect.
Buy I prefer some kind of pseudocode so I can easily make a Mathematica workbook or C++ application. Of course every little tip is very welcome, it doesn't have to be a complete solution!

Many thanks,
Eric

 

[mighty2000]

Hi Eric,

Thank you for sharing your math problem with us. It seems like you have put a lot of thought into creating a fair and efficient tournament algorithm. I would approach this problem by breaking it down into smaller, more manageable parts and then combining them to create a comprehensive solution. Here are some steps that I would take to solve this problem:

1. Determine the minimum number of rounds required for all teams to play every game at least once. This will depend on the total number of teams and games, as well as the number of teams that can play in each game. This will help us establish a baseline for the number of rounds needed in the tournament.

2. Create a table or graph to visualize the teams and games. This will help us see the relationships between teams and games, as well as identify any potential conflicts that may arise based on the rules you have outlined.

3. Use a combination of group theory and graph theory to create an algorithm that assigns teams to games in a fair and efficient manner. This may involve creating a matrix that represents the relationships between teams and games, and then using algorithms such as the Hungarian method or the Ford-Fulkerson algorithm to assign teams to games while following the rules you have outlined.

4. Test and refine the algorithm. This may involve running simulations with different numbers of teams, games, and rounds to see how the algorithm performs. This will also help identify any potential flaws or conflicts that may arise, and allow us to make adjustments to the algorithm.

5. Once the algorithm is finalized, create pseudocode or a flowchart that outlines the steps and logic of the algorithm. This will make it easier to implement the algorithm in a programming language such as Mathematica or C++, as you have mentioned.

I hope this helps guide you in solving your tournament planning problem. Good luck! If you have any further questions or need any clarifications, please don't hesitate to ask.