O(m) Bound on Number of Iterations in Sphere Methods for LP
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Abstract
Consider the linear program (LP): minimize z= cx, subject to Axb, where A is an mn matrix. Sphere methods (SMs) for solving this LP were introduced in Murty (5, 6), eve n though this name was not used there. Theorems in those papers claimed that a version of this method needs at mo st O(m) iterations to solve this LP, however Mirzaian (2) 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 speci al case of the open problem 2 in page 441 of the book Murty (7) dealing only with inspheres encountered in the SM.
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