A linear programming based heuristic for a hard clustering problem on trees
Clustering problems with relational constraints in which the underlying graph is a tree arise in a variety of applications: hierarchical data base paging, communication and distribution network districting, biological taxonomy, and others. They are formulated here as optimal tree partitioning problems. In a previous paper, it was shown that their computational complexity strongly depends on the nature of the objective function and, in particular, that minimizing the total within-cluster dissimilarity or the diameter is computationally hard. We propose a heuristic which finds good partitions for the first problem within a reasonable time, even when its size is large. Such heuristic is based on the solution of a linear program and a maximal network flow one, and in any case it yields an explicit estimate of the relative approximation error. With minor variations a similar approach yields good solutions for the minimum diameter problem.
Key wordsContiguity-constrained clustering
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- Ahuja, R.K. & Magnanti, T.L. & Orlin, J.B. (1993), Network flows, Prentice Hall, New Jersey.Google Scholar
- Garey, M R & Johnson, D.S. (1979). Computers and intractability: a guide to the theory of NP-completeness, Freeman, S.Francisco.Google Scholar
- Hansen, P.& Jaumard, B. & Simeone, B. & Doring V. (1993). Maximum split clustering under connectivity constraints, Cahier G-93-06, GERAD, Montréal.Google Scholar
- Lari, I. & Maravalle, M. & Simeone B. (1998). Linear Programming based heuristics for two hard clustering problems on trees, Technical Report, Dip. di Statistica, Probabilità e Statistiche Applicate, n. 2, 1998.Google Scholar
- Lovász, L. (1979). Combinatorial Problems and Exercises, North Holland, Amsterdam.Google Scholar
- Marcotorchino, J.F. & Michaud, P. (1979). Optimisation en analyse des données, Masson.Google Scholar
- Nemhauser, G.L. & Rinnoy Kan, A.H.G. & Todd M.J. (1989). Handbooks in Operations Research and Management Science, vol. I, Elsevier Science Publishers B.V, Amsterdam.Google Scholar
- Nemhauser, G.L. & Wolsey, L A. (1988). Integer and Combinatorial Optimization, Wiley, New York.Google Scholar