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Asked: 2026-06-14T12:44:05+00:00

Here is my problem scenario: I have a few thousand objects. Each object has

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Here is my problem scenario:

I have a few thousand objects. Each object has 256 Boolean dimensions (true or false). I want to find clusters such that

  1. Each cluster has a minimum amount of true dimensions (a dimension of a cluster is true iff any object in that cluster has this dimension market as true).
  2. The overall sum of all true dimensions over all clusters is minimal.
  3. Each cluster is not bigger than a certain predefined value.

The optimality of the solution is not required, however the algorithm should be fast.

How should I best approach this problem? Is there an algorithm that you would recommend?

Note: I already implemented a brute force approach to this problem, but it is quite slow.

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1 Answer

  1. Editorial Team

    You can write this as a mixed-integer linear program (MILP):

    You have a fixed amount of clusters and objects.
    Each cluster can have at most 256 true dimensions.
    Parameter D_{k,j} is equal to 1 if dimension i is true in object k.

    You have the following variables:

    1. d_{i,j} is a binary variable equal to 1 if dimension j is a true dimension of cluster i.
    2. o_{i,k} is a binary variable that is true if object k is in cluster i.

    You have the following constraints:

    1. Each object can be in only one cluster
    2. A dimension is true in cluster iif it is true in all objects inside cluster
    3. Each cluster can only hold M objects

    The second constraint is a tricky one because it doesn’t feel linear, but actually you can write it linearly.
    The constraints can be written as:

    1. C1 for all k
    2. C2 for all i and j
    3. C3 for all i

    The objective function can be the sum of all d_{i,j}, so you minimize the overall sum of all true dimensions over all clusters.

    Let me explain the second constraint: on the right-hand-side, you compute the number of elements inside cluster i, minus the number of objects having dimension j set to one. This is equal to zero if all objects have dimension j, or something positive if not.

    If this evaluates to zero, then d_{i,j} must be equal to one to avoid violating the constraint. If not, d_{i,j} can be anything (zero or one). This works because d_{i,j} will appear in the objective function, which means that when the program has the choice between zero or one, it will choose zero.

    Once you write this up, you can solve it using a commercial solver (if you have one, they give free licenses to students, in case you are one) or Coin-OR just to name one.

    Just as a reminder: solving MILPs is an NP-complete problem.

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