Algorithm for creating a school timetable
I've been wondering if there are known solutions for algorithm of creating a school timetable. Basically, it's about optimizing "hour-dispersion" (both in teachers and classes case) for given class-subject-teacher associations. We can assume that we have sets of classes, lesson subjects and teachers associated with each other at the input and that timetable should fit between 8AM and 4PM.
I guess that there is probably no accurate algorithm for that, but maybe someone knows a good approximation or hints for developing it.
Answer
This problem is NP-Complete!
In a nutshell one needs to explore all possible combinations to find the list of acceptable solutions. Because of the variations in the circumstances in which the problem appears at various schools (for example: Are there constraints with regards to classrooms?, Are some of the classes split in sub-groups some of the time?, Is this a weekly schedule? etc.) there isn't a well known problem class which corresponds to all the scheduling problems. Maybe, the Knapsack problem has many elements of similarity with these problems at large.
A confirmation that this is both a hard problem and one for which people perennially seek a solution, is to check this (long) list of (mostly commercial) software scheduling tools
Because of the big number of variables involved, the biggest source of which are, typically, the faculty member's desires ;-)..., it is typically impractical to consider enumerating all possible combinations. Instead we need to choose an approach which visits a subset of the problem/solution spaces.
- Genetic Algorithms, cited in another answer is (or, IMHO, seems) well equipped to perform this kind of semi-guided search (The problem being to find a good evaluation function for the candidates to be kept for the next generation)
- Graph Rewriting approaches are also of use with this type of combinatorial optimization problems.
Rather than focusing on particular implementations of an automatic schedule generator program, I'd like to suggest a few strategies which can be applied, at the level of the definition of the problem.
The general rationale is that in most real world scheduling problems, some compromises will be required, not all constraints, expressed and implied: will be satisfied fully. Therefore we help ourselves by:
- Defining and ranking all known constraints
- Reducing the problem space, by manually, providing a set of additional constraints.
This may seem counter-intuitive but for example by providing an initial, partially filled schedule (say roughly 30% of the time-slots), in a way that fully satisfies all constraints, and by considering this partial schedule immutable, we significantly reduce the time/space needed to produce candidate solutions.
Another way additional constraints help is for example "artificially" adding a constraint which prevent teaching some subjects on some days of the week (if this is a weekly schedule...); this type of constraints results in reducing the problem/solution spaces, without, typically, excluding a significant number of good candidates. - Ensuring that some of the constraints of the problem can be quickly computed. This is often associated with the choice of data model used to represent the problem; the idea is to be able to quickly opt-for (or prune-out) some of the options.
- Redefining the problem and allowing some of the constraints to be broken, a few times, (typically towards the end nodes of the graph). The idea here is to either remove some of constraints for filling-in the last few slots in the schedule, or to have the automatic schedule generator program stop shy of completing the whole schedule, instead providing us with a list of a dozen or so plausible candidates. A human is often in a better position to complete the puzzle, as indicated, possibly breaking a few of the contraints, using information which is not typically shared with the automated logic (eg "No mathematics in the afternoon" rule can be broken on occasion for the "advanced math and physics" class; or "It is better to break one of Mr Jones requirements than one of Ms Smith ... ;-) )
In proof-reading this answer , I realize it is quite shy of providing a definite response, but it none the less full of practical suggestions. I hope this help, with what is, after all, a "hard problem".