FPTASs for trimming weighted trees

Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k>=0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weight at mostk and minimize the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard. In this paper, we design an O(nl/@e)-time O(l^2/@e+n)-space algorithm for MaxMstCut, and O(nl(1/@e+logn))-time O(l^2/@e+n)-space algorithms for the other two problems, MinLstCut and MaxLstCut, where n is the number of vertices in the tree, l the number of leaves, and @e>0 the prescribed error bound.

[1]  Oscar H. Ibarra,et al.  Approximation Algorithms for Certain Scheduling Problems , 1978, Math. Oper. Res..

[2]  Mihalis Yannakakis,et al.  The Complexity of Multiterminal Cuts , 1994, SIAM J. Comput..

[3]  Uriel Feige,et al.  Finding small balanced separators , 2006, STOC '06.

[4]  Johannes M. Hattingh,et al.  Tree knapsack approaches for local access network design , 2006, Eur. J. Oper. Res..

[5]  Geon Cho,et al.  The critical‐item, upper bounds, and a branch‐and‐bound algorithm for the tree knapsack problem , 1998 .

[6]  Dániel Marx,et al.  Parameterized graph separation problems , 2004, Theor. Comput. Sci..

[7]  Geon Cho,et al.  A Depth-First Dynamic Programming Algorithm for the Tree Knapsack Problem , 1997, INFORMS J. Comput..

[8]  Robert Krauthgamer,et al.  On Cutting a Few Vertices from a Graph , 2003, Discret. Appl. Math..

[9]  Hiroshi Nagamochi.  Minimum Degree Orderings , 2007, ISAAC.

[10]  Ivan Marsic,et al.  Exact and Heuristic Algorithms for Dynamic Tree Simplification , 2005, J. Math. Model. Algorithms.

[11]  Zhimin Huang,et al.  Bargaining, risk and franchising coordination , 1997, Comput. Oper. Res..

[12]  Geon Cho,et al.  An efficient algorithm for a capacitated subtree of a tree problem in local access telecommunication networks , 1997, Comput. Oper. Res..

[13]  D. S. Johnson,et al.  On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees , 1983, Math. Oper. Res..

[14]  Sartaj Sahni,et al.  General Techniques for Combinatorial Approximation , 1977, Oper. Res..

[15]  Deeparnab Chakrabarty,et al.  Knapsack Problems , 2008 .

[16]  Curt Jones,et al.  Finding Good Approximate Vertex and Edge Partitions is NP-Hard , 1992, Inf. Process. Lett..

[17]  David W. Aha,et al.  Simplifying decision trees: A survey , 1997, The Knowledge Engineering Review.

[18]  Takuro Fukunaga,et al.  FPTAS's for Some Cut Problems in Weighted Trees , 2010, FAW.

[19]  Ojas Parekh,et al.  A Unified Approach to Approximating Partial Covering Problems , 2006, ESA.

[20]  Hiroshi Nagamochi,et al.  Algorithms for the minimum partitioning problems in graphs , 2007 .

[21]  C. Jordan.  Sur les assemblages de lignes. , 1869 .