Abstract
A problem of solvability for the system of equations of the formAx=D|x|+δ is investigated. This problem is proved to beNP-complete even in the case when the number of equations is equal to the number of variables, the matrixA is nonsingular,A≥D≥0,δ≥0, and it is initially known that the system has a finite (possibly zero) number of solutions. For an arbitrary system ofm equations ofn variables, under additional conditions that the matrixD is nonnegative and its rank is one, a polynomial-time algorithm (of the orderO((max{m, n})3)) has been found which allows to determine whether the system is solvable or not and to find one of such solutions in the case of solvability.
Abstract
Изучается задача разрешимости для системы уравнений видаAx=D|x|+δ. Показано, что эта задача являетсяNP-поляой даже в случае, когда число уравнений равно числу переменных, матрипаA невырождена,A≥D≥0,δ≥0 и заранее известно, что аистема имеет конечное (возможно, равное нулю) число решений. Для произвольной системыm уравнений отn переменных при донолнительном условии, что матрицаD не отрицательна и ее ранг равен единице, найден полиномиальный алгоритм (гюрядкаO((max{m, n})3)), позволяющий выяснить разрешимость этой системы и, в случае разрешимости, найти одно из решений.
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© A. V. Lakeyev, 1996
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Lakeyev, A.V. On the computational complexity of the solution of linear systems with moduli. Reliable Comput 2, 125–131 (1996). https://doi.org/10.1007/BF02425914
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DOI: https://doi.org/10.1007/BF02425914