On Half-integrality of Network Synthesis Problem

Network synthesis problem is the problem of constructing a minimum cost network satisfying a given ow-requirement. A classical result of Gomory and Hu is that if the cost is uniform and the ow requirement is integer-valued, then there exists a half-integral optimal solution. They also gave a simple algorithm to nd a half-integral optimal solution. In this paper, we show that this half-integrality and the Gomory-Hu algorithm can be extended to a class of fractional cut-covering problems dened by skew-supermodular functions. Application to approximation algorithm is also given.

[1]  Special skew-supermodular functions and a generalization of Mader ' s splitting-o theorem , 2011 .

[2]  K. Steiglitz,et al.  The Design of Minimum-Cost Survivable Networks , 1969 .

[3]  Bernhard Korte,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[4]  David P. Williamson,et al.  The Design of Approximation Algorithms , 2011 .

[5]  Dieter Jungnickel,et al.  Graphs, Networks, and Algorithms , 1980 .

[6]  Jack Edmonds,et al.  Matching, Euler tours and the Chinese postman , 1973, Math. Program..

[7]  Yair Bartal,et al.  Probabilistic approximation of metric spaces and its algorithmic applications , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[8]  Hiroyoshi Miwa,et al.  NA-EDGE-CONNECTIVITY AUGMENTATION PROBLEMS BY ADDING EDGES( Network Design, Control and Optimization) , 2004 .

[9]  Zoltán Szigeti.  Hypergraph connectivity augmentation , 1999, Math. Program..

[10]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[11]  A. Frank.  Connections in Combinatorial Optimization , 2011 .

[12]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[13]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[14]  Katharina T. Huber,et al.  Basic Phylogenetic Combinatorics , 2011 .

[15]  Hiroshi Nagamochi,et al.  Minimum Augmentation of Edge-Connectivity between Vertices and Sets of Vertices in Undirected Graphs , 2003, Algorithmica.

[16]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

[17]  András Frank.  Augmenting Graphs to Meet Edge-Connectivity Requirements , 1992, SIAM J. Discret. Math..

[18]  N. Saitou,et al.  The neighbor-joining method: a new method for reconstructing phylogenetic trees. , 1987, Molecular biology and evolution.

[19]  Kamal Jain,et al.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[20]  藤重 悟.  Submodular functions and optimization , 1991 .

[21]  Andrew V. Goldberg,et al.  Improved approximation algorithms for network design problems , 1994, SODA '94.

[22]  Zeev Nutov.  Approximating connectivity augmentation problems , 2005, SODA '05.

[23]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[24]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[25]  T. C. Hu,et al.  An Application of Generalized Linear Programming to Network Flows , 1962 .

[26]  Satish Rao,et al.  A tight bound on approximating arbitrary metrics by tree metrics , 2003, STOC '03.

[27]  William Fahle,et al.  Review of "Graphs, Networks, and Algorithms (second edition) by Dieter Jungnickel", Springer-Verlag 2005 , 2005, SIGA.