On the closest point to the origin in transportation polytopes

We consider the problem of finding the point in the transportation polytope which is closest to the origin. Recursive formulas to solve it are provided, explaining how they arise from geometric considerations, via projections, and we derive solution algorithms with linear computational complexity in the number of variables.

[1]  Leen Stougie,et al.  A Linear Bound On The Diameter Of The Transportation Polytope* , 2006, Comb..

[2]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part II: Convex quadratic programming , 1989, Math. Program..

[3]  Michel Balinski,et al.  Signature classes of transportation polytopes , 1993, Math. Program..

[4]  D. Romero.  Easy transportation-like problems onK-dimensional arrays , 1990 .

[5]  Kazuyuki Aihara,et al.  Size-constrained Submodular Minimization through Minimum Norm Base , 2011, ICML.

[6]  Shmuel Onn,et al.  Permutohedra and minimal matrices , 2006 .

[7]  B. Korte,et al.  Minimum norm problems over transportation polytopes , 1980 .

[8]  Pieter M. Kroonenberg,et al.  A survey of algorithms for exact distributions of test statistics in r × c contingency tables with fixed margins , 1985 .

[9]  Alexander Barvinok,et al.  Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes , 2007, 0709.3810.

[10]  Edward D. Kim,et al.  Combinatorics and geometry of transportation polytopes: An update , 2013, Discrete Geometry and Algebraic Combinatorics.

[11]  Bingsheng He,et al.  Solution of projection problems over polytopes , 1992 .

[12]  Stavros A. Zenios,et al.  A Comparative Study of Algorithms for Matrix Balancing , 1990, Oper. Res..

[13]  Igor Pak,et al.  On the Number of Faces of Certain Transportation Polytopes , 2000, Eur. J. Comb..

[14]  P. Wolfe.  Algorithm for a least-distance programming problem , 1974 .

[15]  L. Khachiyan,et al.  The polynomial solvability of convex quadratic programming , 1980 .

[16]  Jesús A. De Loera,et al.  Graphs of transportation polytopes , 2007, J. Comb. Theory, Ser. A.