An Algorithmic Theory of Integer Programming

Abstract

We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We show that integer programming can be solved in time д(a, d)poly(n, L), where д is some computable function of the parameters a and d, and L is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by a and d, and is solvable in polynomial time for every fixed a and d. Our results also extend to nonlinear separable convex objective functions. Moreover, for linear objectives, we derive a strongly-polynomial algorithm, that is, with running time д(a, d)poly(n), independent of the rest of the input data.We obtain these results by developing an algorithmic framework based on the idea of iterative augmentation: starting from an initial feasible solution, we show how to quickly find augmenting steps which rapidly converge to an optimum. A central notion in this framework is the Graver basis of the matrix A, which constitutes a set of fundamental augmenting steps. The iterative augmentation idea is then enhanced via the use of other techniques such as new and improved bounds on the Graver basis, rapid solution of integer programs with bounded variables, proximity theorems and a new proximity-scaling algorithm, the notion of a reduced objective function, and others.As a consequence of our work, we advance the state of the art of solving block-structured integer programs. In particular, we develop near-linear time algorithms for n-fold, tree-fold, and 2-stage stochastic integer programs. We also discuss some of the many applications of these classes.

A Colorful Steinitz Lemma with Applications to Block Integer Programs

“…Following the work of Eisenbrand and Weismantel [15], the Steinitz Lemma has been used in numerous projects; we point to [8,10,13,14,22,23] just to name a few. One area that benefits from the Steinitz Lemma is the study of integer programs with special sparse block-structures.…”

“…One area that benefits from the Steinitz Lemma is the study of integer programs with special sparse block-structures. We refer to [8,14,19] for more on how the special sparsity structure of block-structured matrices can be used to derive faster algorithms for integer programs. Block-structured integer programs can be applied in various problems such as scheduling and social choice; see, e.g., [21,25,30].…”

“…, t 0 + nt}. Graver bases are often used to solve blockstructured integer programs [8,14,18,19,23]. For n-fold matrices, a Graver basis element g satisfies g 1 ∈ O(s 0 s∆) (s0+1)(s+1) [13, ii) Theorem 5].…”

“…Set v D := −D −1 B i h ℓ for each D ⊆ A i that is a feasible basis matrix for Q; thus, v D ≥ 0. The points v D are the vertices of Q, and they are integer-valued because we have scaled h ℓ to lie in γ • Z t0 and γ satisfies (14). For each D, we use Hadamard's inequality and the definition of ω 0 in (15) to conclude…”

Complexity of optimizing over the integers

“…Listing all the literature here is impossible. Instead, we point to [17,36,44,49,79,88] and the references therein as excellent summaries and starting points for exploring this diverse field of research activity.…”