A primal-simplex based Tardos' algorithm

In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin's primal-based modification and Mizuno's use of the simplex method, we introduce a modification of Tardos' algorithm considering only the primal problem and using simplex method to solve the auxiliary problems. The proposed algorithm is strongly polynomial if the coefficient matrix is totally unimodular and the auxiliary problems are non-degenerate.Comment: 7 page

General Relativistic MHD Simulations of the Gravitational Collapse of a Rotating Star with Magnetic Field as a Model of Gamma-Ray Bursts

We have performed 2.5-dimensional general relativistic magnetohydrodynamic (MHD) simulations of the gravitational collapse of a magnetized rotating massive star as a model of gamma ray bursts (GRBs). This simulation showed the formation of a disk-like structure and the generation of a jet-like outflow inside the shock wave launched at the core bounce. We have found the jet is accelerated by the magnetic pressure and the centrifugal force and is collimated by the pinching force of the toroidal magnetic field amplified by the rotation and the effect of geometry of the poloidal magnetic field. The maximum velocity of the jet is mildly relativistic ($\sim$ 0.3 c).Comment: 4 pages, 1 figure, aipTEX, contribution to the 2003 GRB Conference, held at Santa Fe, N

A Bound for the Number of Different Basic Solutions Generated by the Simplex Method

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the primal problem is nondegenerate, it becomes a bound for the number of iterations. We show some basic results when it is applied to special linear programming problems. The results include strongly polynomiality of the simplex method for Markov Decision Problem by Ye and utilize its analysis.Comment: Keywords: Simplex method, Linear programming, Iteration bound, Strong polynomiality, Basic feasible solution

A Bound for the Number of Different Basic Solutions Generated by the Simplex Method

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the primal problem is nondegenerate, it becomes a bound for the number of iterations. We show some basic results when it is applied to special linear programming problems. The results include strongly polynomiality of the simplex method for Markov Decision Problem by Ye and utilize its analysis.Comment: Keywords: Simplex method, Linear programming, Iteration bound, Strong polynomiality, Basic feasible solution