Polynomial algorithms for partitioning a tree into single-center subtrees to minimize flat service costs

This paper deals with the following graph partitioning problem. Consider a connected graph with n nodes, p of which are centers, while the remaining ones are units. For each unit-center pair there is a fixed service cost and the goal is to find a partition into connected components such that each component contains only one center and the total service cost is minimum. This problem is known to be NP-hard on general graphs, and here we show that it remains such even if the service cost is monotone and the graph is bipartite. However, in this paper we derive some polynomial time algorithms for trees. For this class of graphs we provide several reformulations of the problem as integer linear programs proving the integrality of the corresponding polyhedra. As a consequence, the tree partitioning problem can be solved in polynomial time either by linear programming or by suitable convex nondifferentiable optimization algorithms. Moreover, we develop a dynamic programming algorithm, whose recursion is based on sequences of minimum weight closure problems, which solves the problem on trees in O(np) time. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

[1]  Yair Bartal,et al.  On approximating arbitrary metrices by tree metrics , 1998, STOC '98.

[2]  Sudipto Guha,et al.  Rounding via Trees : Deterministic Approximation Algorithms forGroup , 1998 .

[3]  Antonio Frangioni,et al.  Generalized Bundle Methods , 2002, SIAM J. Optim..

[4]  G. Nemhauser,et al.  Optimal Political Districting by Implicit Enumeration Techniques , 1970 .

[5]  P. Hammer,et al.  Order Relations of Variables in 0-1 Programming , 1987 .

[6]  Andreas Griewank,et al.  On constrained optimization by adjoint based quasi-Newton methods , 2002, Optim. Methods Softw..

[7]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[8]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[9]  Bruno Simeone,et al.  Evaluation and Optimization of Electoral Systems , 1987 .

[10]  O. Kariv,et al.  An Algorithmic Approach to Network Location Problems. II: The p-Medians , 1979 .

[11]  Bruno Simeone,et al.  A spanning tree heuristic for regional clustering , 1995 .

[12]  Jean-Philippe Vial,et al.  Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method , 2002, Optim. Methods Softw..

[13]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[14]  C. A. Rogers,et al.  Packing and Covering , 1964 .

[15]  Gerard Cornuejols Combinatorial optimization , 1987 .

[16]  G. Cornuéjols,et al.  Combinatorial optimization : packing and covering , 2001 .