A polynomial oracle-time algorithm for convex integer minimization

TL;DR

This paper introduces a polynomial-time algorithm for solving convex integer minimization problems using greedy augmentation with Graver bases, extending to stochastic and N-fold cases, with practical applications.

Contribution

It presents a novel polynomial oracle-time algorithm for convex integer minimization leveraging Graver bases, extending to stochastic and N-fold problems.

Findings

Polynomially many augmentation steps needed for certain convex integer problems

Extension of results to convex N-fold and 2-stage stochastic integer minimization

Applications demonstrating polynomial time solutions for specific convex N-fold problems

Abstract

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.