O(m) Bound on Number of Iterations in Sphere Methods for LP
Consider the linear program (LP): minimize z = cx, subject to Ax ≥ b, where A is an m × n matrix. Sphere methods (SMs) for solving this LP were introduced in Murty [5, 6], even though this name was not used there. Theorems in those papers claimed that a version of this method needs at most O(m) iterations to solve this LP, however Mirzaian  pointed out an error in the proofs of these theorems there. Here we prove the claim using the geometry of inspheres. Also the results in this paper provide a solution to the special case of the open problem 2 in page 441 of the book Murty  dealing only with inspheres encountered in the SM.