N-fold integer programming in cubic time

N-fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer programming. The fastest algorithm for $n$-fold integer programming predating the present article runs in time $O(n^{g(A)}L)$ with $L$ the binary length of the numerical part of the input and $g(A)$ the so-called Graver complexity of the bimatrix $A$ defining the system. In this article we provide a drastic improvement and establish an algorithm which runs in time $O(n^3 L)$ having cubic dependency on $n$ regardless of the bimatrix $A$. Our algorithm can be extended to separable convex piecewise affine objectives as well, and also to systems defined by bimatrices with variable entries. Moreover, it can be used to define a hierarchy of approximations for any integer programming problem

Characterization of matrices with bounded graver bases and depth parameters and applications to integer programming

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the \u1d4c1₁-norm of the Graver basis is bounded by a function of the maximum \u1d4c1₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the \u1d4c1₁-norm of the Graver basis of the constraint matrix, when parameterized by the \u1d4c1₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix