1. Yuzhong Qu Nanjing University
Consensus partition
Yuzhong Qu
Nanjing University

2. Problem W is a set of properties At some moments, a user might
It has a public alignment, e.g. an equivalence relation R on W.
An equivalence relation can also be represented by a partition
At some moments, a user might
See a list of a subset of W
Align elements of W (move/remove an item to/from a partition)
The problem
How to preserve personal alignment for each user?
How to improve (optimize) the public alignment according to users’ alignments?

3. Global alignment (Representation)
Global alignment: an equivalence relation on W
A function from W to {0}N (c: W{0}N)

4. User’s alignment (Representation)
A (partial) function from Wi to {0, i}N (ci: Wi {0, i}N)
Personalized alignment
ci  c|W-Wi : W{0, i}N

5. Optimizing the global alignment
Input: a family of users’ partitions {ci}
Output: an optimal partition c* on Wi
min id(c*, ci)
Bn=j=1..n S(n,j)

6. Partition distance
Vertex cover problem

7. Consensus Partition Clique partitioning problem
 having a maximum total weight

8. Reference
Gusfield D. Partition-distance: A problem and class of perfect graphs arising in clustering[J]. Information Processing Letters, 2002, 82(3):
Guénoche A. Consensus of partitions: a constructive approach[J]. Advances in Data Analysis and Classification, 2011, 5(3):

9. Websoft Research Group
Acknowledgement
Websoft Research Group