## Consensus of partitions: a constructive approach.(English)Zbl 1253.68258

Summary: Given a profile (family) $$\Pi$$ of partitions of a set of objects or items $$X$$, we try to establish a consensus partition containing a maximum number of joined or separated pairs in $$X$$ that are also joined or separated in the profile. To do so, we define a score function, $$S_\Pi$$ associated to any partition on $$X$$. Consensus partitions for $$\Pi$$ are those maximizing this function. Therefore, these consensus partitions have the median property for the profile and the symmetric difference distance. This optimization problem can be solved, in certain cases, by integer linear programming. We define a polynomial heuristic which can be applied to partitions on a large set of items. In cases where an optimal solution can be computed, we show that the partitions built by this algorithm are very close to the optimum which is reached in practically all the cases, except for some sets of bipartitions.

### MSC:

 68R05 Combinatorics in computer science 90C27 Combinatorial optimization 68W25 Approximation algorithms

### Keywords:

partitions; consensus; transfer distance; simulation
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### References:

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