On the strictly complementary slackness relation in linear programming

Balinski and Tucker introduced in 1969 a special form of optimal tableaus for LP, which can be used to construct primal and dual optimal solutions such that the complementary slackness relation holds strictly. In this paper, first we note that using a polynomial time algorithm for LP Balinski and Tucker’s tableaus are obtainable in polynomial time. Furthermore, we show that, given a pair of primal and dual optimal solutions satisfying the complementary slackness relation strictly, it is possible to find a Balinski and Tucker’s optimal tableau in strongly polynomial time. This establishes the equivalence between Balinski and Tucker’s format of optimal tableaus and a pair of primal and dual solutions to satisfy the complementary slackness relation strictly. The new algorithm is related to Megiddo’s strongly polynomial algorithm that finds an optimal tableau based on a pair of primal and dual optimal solutions. Key words: Linear programming, the simplex tableau, primal and dual solu-tions, complementary slackness, strongly polynomial-time