Polynomial algorithms for LP over a subring of the algebraic integers with applications to LP with circulant matrices

It is shown that a modified variant of the interior point method can solve linear programs (LPs) whose coefficients are real numbers from a subring of the algebraic integers. By defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring, it is proved that the modified algorithm runs in time polynomial in the encoding size of the input coefficients, the dimension of the problem, and the order of the subring. The Tardos scheme is then extended to this case, yielding a running time that is independent of the objective and right-hand side data. As a consequence of these results, it is shown that LPs with real circulant coefficient matrices can be solved in strongly polynomial time. It is also shown how the algorithm can be applied to LPs whose coefficients belong to the extension of the integers by a fixed set of square roots