Question

I am trying to implement an app which assigns s students to l labs in g lab groups. The constraints are:

1:students shall work with new students for every lab.
2:all students shall be lab leader once.

2 is not solvable if the students can’t be divided evenly in the lab groups. Therfore it is acceptable if the “odd” students never get to be lab leader.

I have tried two approaches but I am not happy yet.:

1. Tabu search, which solves 1 but has problems solving 2 ( I actually first solve 1 and then try to solve 2, which might be the wrong approach, any suggestions)

2. A simple solution where I divide the students in the #labs in an array [0..6][7..14][15..21] and then rotate(with 0,1,2 inc) and transpose the matrix, repeat this for #labs times with incremented rotation (1,2,4) and (2,4,6). For 21 students in 3 labs with lab groups of 7 the result looks like this:

   • lab 1: [0, 7, 14], [1, 8, 15], [2, 9, 16], [3, 10, 17], [4, 11, 18], [5, 12, 19], [6, 13, 20]
   • lab 2: [6,12, 18], [0, 13, 19], [1, 7, 20], [2, 8, 14], [3, 9, 15], [4, 10, 16], [5, 11, 17]
   • lab 3: [5, 10, 15], [6, 11, 16], [0, 12, 17], [1, 13, 18], [2, 7, 19], [3, 8, 20], [4, 9, 14]

the lab leaders are the first column for lab 1, the second for lab 2 …

This solution works decent but for instance fails for 12 students in 3 labs or 150 students in 6 labs. Any suggestions?

2 seems to handle the same number of cases or combinations, and is lightning fast compared to 1. Maybe I should get a noble price 🙂

Answer 1

Constraint #1 alone is usually referred to as the social golfer problem. (Let parameters g be the number of groups and s be the size of each group and w be the number of weeks. A grouping is a partition of g * s golfers into g groups of size s. Determine whether w groupings can be found such that each pair of golfers are grouped together at most once.) The social golfer problem has been studied in the combinatorial optimization literature, and the approaches are of three types (you can use your favorite search engine to find the research articles):

1. Local search. This is effective when w is well below its maximum feasible value. Dotú and Van Hentenryck have a paper applying tabu search to the social golfer problem.

2. Complete search. This is necessary when w is above or just below its maximum feasible value but it does not scale very well.

3. Algebraic constructions. This is how the notorious g=8 s=4 w=10 instance was solved. Unfortunately, for many parameter sets there is no construction known.

To assign lab leaders, compute a maximum matching between students and lab groups, where there is an edge between a student and a lab group if that student belongs to that lab group.