n-Fold integer programming in cubic time

Abstract

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.

Covering a tree with rooted subtrees – parameterized and approximation algorithms

“…To achieve the FPT algorithm, we first formulate the problem as an integer linear program having a certain "tree-fold" structure. Then we show that an ILP with such a structure is FPT, which is a generalization of an earlier FPT result for n-fold integer programming by Hemmecke, Onn and Romanchuk [5]. This extension of n-fold ILP may be of independent interest.…”

“…This statement remains true if, instead of choosing the best possible augmentation vector of the form γg, say, γ * g * , we choose an augmentation vector q which is at least as good as γ * g * . That is, if in each augmentation vector we choose an augmentation vector q such that c T q ≤ γ * c T g * , the optimal solution x * could also be achieved after O(nL) augmentation vectors [2,5]. Notice that q does not necessarily belong to G(A).…”

“…Very recently, Knop and Koutecký [13] observes that many scheduling problems can be formulated as an integer program with a special structure called n-fold integer program. Exploiting the FPT algorithm for the n-fold integer program [5] they are able to show an FPT algorithm for various scheduling problems.…”

“…Indeed, our FPT algorithm relies on an FPT algorithm for a more general integer programming problem, which extends the existing FPT algorithm for the n-fold integer programming [5]. We consider the following integer programming:…”

“…Indeed, they show that a variety of scheduling problems, including P ||C max , could be formulated as an n-fold integer programming. Applying the FPT algorithm for n-fold integer programming by Hemmecke, Onn and Romanchuk [5], an FPT algorithm for P ||C max follows. It is worth mentioning that parameterized studies for integer programming that has a sparse structure have received much attention in the literature, e.g., [9,14,15].…”