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

TL;DR

This paper introduces a polynomial-time algorithm for a broad class of N-fold 4-block decomposable integer programs, enabling efficient solutions for fixed block structures and variable scenarios or objectives.

Contribution

It generalizes existing N-fold and two-stage integer programs to a more comprehensive class and provides polynomial-time solvability and optimality certificates for these models.

Findings

Polynomial-time solvability for fixed blocks and variable N.

Efficient solutions for stochastic integer programs with complex constraints.

Application to multi-commodity flow problems with polynomial complexity.

Abstract

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.