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A linear programming algorithm is called genuinely polynomial if it requires no more than p (m, n) arithmetic operations to solve problems of order m n, where p is a polynomial. It is not known whether such an algorithm exists. We present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality. The number of operations required is O (mn³ m). The technique used was developed in a previous paper where a novel binary search idea was introduced.
Nimrod Megiddo (Sun,) studied this question.