A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex $N$-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex $N$-fold integer minimization problems for which our approach provides polynomial time solution algorithms.Comment: 19 pages, 1 figur

A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions

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