# The clustering matroid and the optimal clustering tree

- First Online:

- Received:
- Accepted:

- 7 Citations
- 178 Downloads

## Abstract.

We consider the following problem: Given a complete graph *G*=(*V*,*E*) with a cost on every edge and a given collection of subsets of *V*, we have to find a minimum cost spanning tree *T* such that each subset of the vertices in the collection induces a subtree in *T*. One motivation for this problem is to construct a minimum cost communication tree network for a collection of non-disjoint groups of customers such that the network will provide ``group fault tolerance'' and ``group privacy''. We model this problem as a matroid. We extend it to general matroids and call the new matroids ``clustering matroids''. We define three variations of the clustering tree problem and show that from an algorithmic point of view they are polynomially equivalent. We present a polynomial algorithm for one of the three variations, which implies that all of them can be solved polynomially. For the case where the cardinality of the subsets in the collection does not exceed three, we provide a greedy algorithm, a linear algorithm and also a polyhedron description of the convex hull of all the feasible solutions.

### Keywords

combinatorial optimization clustering spanning trees matroids hypergraphs polynomial graph algorithms primal-dual algorithms polyhedra## Preview

Unable to display preview. Download preview PDF.

### References

- 1.Bixby, R.E.: Recent Algorithms for Two Versions of Graph Realization and Remarks on Applications to Linear Programming, Progress in Combinatorial Optimization, W.R. Pulleyblank (Ed.), Academic Press Canada 1984, pp. 39–67Google Scholar
- 2.Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, American Elsevier Publishing, New York, 1976Google Scholar
- 3.Booth, K.S., Leuker, G.S.: Testing for the Consecutive Ones Property, Interval Graph and Graph Planarity Using PQ-Tree Algorithms. J. Comp. Sys. Sci.
**13**, 335–379 (1976)Google Scholar - 4.Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. John Wiley and Sons, New York, 1998Google Scholar
- 5.Edmonds, J.: Matroids, Submodular Functions, and Certain Polyhedra in Combinatorial Structures and their Applications (R.K. Guy, H. Hanani, N. Sauer, J. Schonheim, eds.), Gordon and Breach, New York, 1970 pp. 69–87Google Scholar
- 6.Edmonds, J.: Matroids and the Greedy Algorithm. Mathematical Programming
**1**, 127–136 (1971)Google Scholar - 7.Garey, M.R., Johnson, D.S.: Computers and Intractability, A guide to the theory of NP-Completeness. W.H. Freeman and Company, San Francisco, 1979Google Scholar
- 8.Gavril, F.: A Recognition Algorithm for the Intersection Graphs of Paths in Trees. Discrete Mathematics
**23**, 211–227 (1978)Google Scholar - 9.Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980Google Scholar
- 10.Golumbic, M.C., Jamison, R.E.: Edge and Vertex Intersection of Paths in a Tree. Discrete Mathematics, North-Holland
**55**, 151–159 (1985)Google Scholar - 11.Korach, E., Stern, M.: The Clustering TSP Problem. In preparationGoogle Scholar
- 12.McKee, T.A., McMorris, F.R.: Topics in Intersection Graph Theory. SIAM Monographs on Discrete Mathematics and Applications, 1999Google Scholar
- 13.Rado, R.: A Note on Independence Functions. Proceedings of the London Mathematical Society
**7**, 300–320 (1957)Google Scholar - 14.Swaminathan, R., Wagner, D.K.: On the Consecutive-Retrieval Problem. SIAM J. Comp.
**23**(2), 398–414 (1994)Google Scholar - 15.Welsh, D.J.A.: Matroid Theory. Academic Press, London, 1976Google Scholar