Arranging a 2D array of matchups to ensure an equal distribution of home and away matches for each team

I am in the process of developing a unique UEFA Champions League 24 'Swiss Model' tournament with 36 teams. Each team is set to compete against 8 different opponents, resulting in a total of 144 matches. I already have a list of matchups prepared. Is there a method to organize these matchups so that every team can have an equal number (in this case, 4) of home and away games? The first position in each matchup designates the home team, while the second position indicates the away team.

I've made an attempt to verify if the total number of home games for the first team exceeds 4 matches (given the 8 rounds). If it does, then it switches to playing as the away team. However, the results are still not evenly distributed.

Example on JSFiddle

const matchups = [
    [
        "35",
        "36"
    ],
    ...
]

// JavaScript code continues here...
javascripttypescriptalgorithmsorting

Answer №1

To solve this problem, utilize the max-flow-min-cut method.

Construct the network in the following manner:

Nodes: Beginning Point, Endpoint, 1 for each team, 1 for each match.

Connections:

  • start -> team_i, limit 4, for every team.
  • team_i -> match_j, limit 1, for all 8 matches that involve team_i.
  • match_j -> end, limit 1, for each individual match.

Determine the maximum flow in this network using the max-flow-min-cut approach. Since all capacities are whole numbers, an integer solution can be achieved (where all flows are integers). Think of a flow from a team to a match as indicating that the team is playing at home in that particular match.

If there is a feasible answer, it results in a total flow of 144, equating to 1 for each match, and this value can be identified by the max-flow-min-cut algorithm.

Answer №2

Let's break down a scenario involving a 5-team pool:

Imagine a schedule for home and away matches featuring teams labeled A through E:

home\away A B C D E
A - O O X X
B X - O O X
C X X - O O
D O X X - O
E O O X X -

This arrangement indicates, for example, that Team A will host Team B and C while visiting Teams D and E.

I have devised the home/away match matrix without strict rules: ensuring each team has 2 home games and 2 away games (in this specific 5-team setting).

If you introduce some randomness when pairing your array of [Team 1, Team 2, ...] with the list of [Team A, Team B, ...], you'll create an unpredictable championship format with equal opportunities for all teams.

Please let me know if further clarification is needed.

UPDATE:

It's unclear whether all 9 teams in every pool must face one another once or if the 36 total teams should contend against 8 random opponents to avoid repeat matchups.

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