constraint satisfaction problem missing one constraint

Question

I'm a lab practises tutor at the university, based on last year student comments, we wanted, my boss and I, to address them. My boss chose to go with writing a C script and I pick python (python-constraint) to try to resolve our problem.

Informations

• There are 6 sessions
• There are 4 roles
• There are 6 practices
• There are 32 students
• There are 4 students per team

Problem :

Assign each student to 4 roles, in 4 practices in 4 different sessions.

Constraints :

1. Students should do a role once
2. Students should do 4 different practices out of 6
3. Students should do only one practice per session
4. Student should meet the same mate only once

Templates :

Here is the template that I feel with students, where each team is composed of 4 students, positions [0, 1, 2 or 3] are roles assigned to them. Each available position is numbering from 1 to 128

[# Semester
   [ # Session
     [ # Practice/Team
1, 2, 3, 4],
  [5, 6, 7, 8],
  [9, 10, 11, 12],
  [13, 14, 15, 16],
  [17, 18, 19, 20],
  [21, 22, 23, 24]],
 [[25, 26, 27, 28],
  [29, 30, 31, 32],
  [33, 34, 35, 36],
  [37, 38, 39, 40],
  [41, 42, 43, 44],
  [45, 46, 47, 48]],
 [[49, 50, 51, 52],
  [53, 54, 55, 56],
  [57, 58, 59, 60],
  [61, 62, 63, 64],
  [65, 66, 67, 68],
  [69, 70, 71, 72]],
 [[73, 74, 75, 76],
  [77, 78, 79, 80],
  [81, 82, 83, 84],
  [85, 86, 87, 88],
  [89, 90, 91, 92],
  [93, 94, 95, 96]],
 [[97, 98, 99, 100],
  [101, 102, 103, 104],
  [105, 106, 107, 108],
  [109, 110, 111, 112]],
 [[113, 114, 115, 116],
  [117, 118, 119, 120],
  [121, 122, 123, 124],
  [125, 126, 127, 128]]]

In other words :

This is a session :

 [[1, 2, 3, 4],
  [5, 6, 7, 8],
  [9, 10, 11, 12],
  [13, 14, 15, 16],
  [17, 18, 19, 20],
  [21, 22, 23, 24]],

Those team do the same practice:

[
    [1, 2, 3, 4],
    [25, 26, 27, 28],
    [49, 50, 51, 52],
    [73, 74, 75, 76],
    [97, 98, 99, 100],
    [113, 114, 115, 116]
]

Those position do the same role :

[
   1,
   5,
   9,
   13,
   17,
   21,
   25,
   ...
]

What I have so far :

Using python-constraint I was able to validate the first three constraints :

Valid solution : False
            - sessions  : [True, True, True, True, True, True]
            - practices : [True, True, True, True, True, True]
            - roles     : [True, True, True, True]
            - teams     : [False, False, True, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, False, True, False, False, False, False, False]

For those that may interesting I simply do like this :

For each condition I use AllDifferentConstraint. For example, for one session I do:

problem.addConstraint(AllDifferentConstraint(), [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24])

I'm not able to find a way to constraint team, my last attempt on the entire semester was this :

    def team_constraint(self, *semester):
        students = defaultdict(list)

        # get back each teams based on the format [# Semester [ #Session [# Practice/Team ... 
        teams = [list(semester[i:i+4]) for i in range(0, len(semester), 4)]

        # Update Students dict with all mate they work with
        for team in teams:
            for student in team:
                students[student] += [s for s in team if s != student]

        # Compute for each student if they meet someone more than once 
        dupli = []
        for student, mate in students.items():
            dupli.append(len(mate) - len(set(mate)))

        # Loosly constraint, if a student meet somone 0 or one time it's find
        if max(dupli) >= 2:
            print("Mate encounter more than one time", dupli, min(dupli) ,max(dupli))
            return False
        pprint(students)
        return True

Questions :

1. Is it possible to do what I want for the team conditions ? What I mean is I have no idea if it is possible to assign 12 mates to each student and each of them meet the same mate only once.
2. For the team constraint, did I miss a more performant algorithm ?
3. Any pist that I can follow ?

Answer 1

The main question would be answered with something like...

   def person_works_with_different():
        # over all the sessions, each person works with each other person no more than once.
        # 'works with' means in 'same session team'
        for p in all_people:
            buddy_constraint = []
            for s in all_sessions:
                for g in all_teams:
                    p_list = [pv[k] for k in filter(lambda i: i[P] == p and i[S] == s and i[G] == g, pv)]
                    for o in all_people:
                        if o != p:  # other is not person
                            o_list = [self.pv[k] for k in filter(lambda i: i[self.P] == o and i[self.S] == s and i[self.G] == g, self.pv)]
                            tmp = model.NewBoolVar('')
                            buddy_constraint.append(tmp)
                            model.Add(sum(o_list) == sum(p_list)).OnlyEnforceIf(tmp)
                            # tmp is set only if o and p are in the same session/team
            # The number of times a student gets to take part is the number of roles.
            # The size of the group controlled by the number of roles
            model.Add(sum(buddy_constraint) = all_roles * (all_roles - 1)) 

Added Edit

I had another look at your problem yesterday - (admittedly not long, as I have a lot of work on at the moment), and...

First of all, I see that your 'team' entity, is pretty much what I called an 'action' entity, and in retrospect I think 'team' (or 'group') was a better word for it.

If you are still finding the constraints hard, I suggest you break them out, and work on them individually - particularly the team/person/session constraints, followed by the role/task constraints.

/Added Edit

team: a gathering of 4 persons during a session
person (32): a participant of a team
session (6): time: eg, 8am -10am
role (4): what responsibility a person has in an action
task (6): type of action

A person does:
 0..1 action per session-group
 1 role per action
 1 task per action
 0..1 of each task
 1 of each role in an action
 4 persons in an action

A person meets each other person 0..1 times
An action requires exactly 4 people

I had a similar problem recently, and in the end turned to OR-tools. https://developers.google.com/optimization/cp/cp_solver

Particularly, have a look at the nurse scheduling problem: https://developers.google.com/optimization/scheduling/employee_scheduling#nurse_scheduling

Anyhow, the problem is not too complex, so maybe using a solver would be overkill for you.

Likewise, for this sort of problem it may be better to use a tuple-keyed dict to hold your variables, rather than nested lists:

{ Team, Session, Person: BoolVar }

The main reason is that you can then apply constraints via filters, which is much easier than having to do nested list manipulations, for instance, to apply a constraint across persons/teams, you can do (where person is index 2 and team is index 0):

for p in all_persons:
    for t in all_teams:
        stuff = [b_vars[k] for k in filter(lambda i: i[2] == p and i[0] == t, b_vars)]
        model.Add(sum(stuff) == 4)  # persons per team == 4