A polynomial projection-type algorithm for linear programming

or show that none exists. The first practical algorithm for linear programming was the simplex method, introduced by Dantzig in 1947 [7]; while efficient in practice, for most known pivoting rules the method has an exponential-time worst case complexity. Several other algorithms were developed over the subsequent decades, such as the relaxation method by Agmon [2] and Motzikin and Shoenberg [13]. The first polynomial-time algorithm, the ellipsoid method, was introduced by Khachiyan [11], followed a few years later by Karamarkar’s first interior point method [10]. In 2010 Chubanov [4, 5] gave a different type of polynomial time algorithm, inspired by the relaxation method, followed recently by a substantially simpler and improved algorithm [6]. Computational experiments of Chubanov’s original algorithm, as well as a different treatment, were carried out by Basu, De Loera and Junod [3]. Here we present a polynomial time algorithm based on [4]. The engine behind our algorithm is the Bubble algorithm subroutine, which can be considered as an unfolding of the recursion in the Divide-andConquer algorithm described in the earlier paper of Chubanov [4]. Our algorithm is also related to the one in [6]; in particular, our Bubble algorithm is an analogue of the Basic algorithm in [6]. However, while our Bubble algorithm is a variant of the relaxation method1, Chubanov’s Basic algorithm is precisely von Neumann’s algorithm (see Dantzig [8]). The two algorithms proceed in a somewhat different manner. Chubanov’s algorithm decides whether Ax = 0 has a strictly positive solution, and reduces problems of the form (1) via an homogenization, whereas we work directly with the form (1). Also, the key updating step of the bounds on the feasibility region after an iteration of the basic subroutine and the supporting argument substantially differs from ours. In particular, whereas [6] divides only one of the upper bounds on the variables by exactly two, our algorithm uses simultaneous updates of multiple components. Another difference is that instead of repeatedly changing the original system by a rescaling, we keep the same problem setting during the entire algorithm and modify a certain norm instead. This enables a clean understanding of the progress made by the algorithm. If we denote by L the encoding size of the matrix (A, b), our algorithm performs O([n5/ log n]L) arithmetic operations. Chubanov’s algorithm [6] has a better running time bound of O(n4L); however, note that our algorithm is still a considerable improvement over O(n18+3ǫL12+2ǫ) in the previous version [5]. We get a better boundO([n/ log n]L) on the number of executions of the basic subroutine, as compared