A Polynomial-Time Algorithm for Optimizing over N-Fold 4-Block Decomposable Integer Programs

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.

Graver basis and proximity techniques for block-structured separable convex integer minimization problems

“…The above result has appeared before in [8]; we included the proof to make the present paper more self-contained. We now complement it with a useful alternative bound, which is given by the following new result.…”

“…The paper [8] and the present paper crucially rely on the following structural result about G ( C D B A ) (N ) , which was proved in [8].…”

n-Fold integer programming in cubic time

“…For more information see e.g. [4,5,7,9,10,11,12,14,15,20], [17,Chapters 4,5], and the references therein.…”

Integer Programming and Incidence Treedepth

“…Similarly, if A 1 and A 2 are void, we obtain the 2-stage stochastic IP. The rst, up to our knowledge, pioneering algorithmic work on n-fold 4-block IPs is due to Hemmecke et al [13]. They gave an algorithm that given n, the 2 × 2 block matrix E, and vectors w, b, l, u nds an integral vector x with E (n) x = b, l x u minimizing wx.…”

“…They gave an algorithm that given n, the 2 × 2 block matrix E, and vectors w, b, l, u nds an integral vector x with E (n) x = b, l x u minimizing wx. The algorithm of Hemmecke et al [13] runs in time n g(r,s, E ∞) L, where r is the number of rows of E, s is the number of columns of E, L is the size of the input, and g : N → N is a computable function. Thus, from the parameterized complexity viewpoint this is an XP algorithm for parameters r, s, E ∞ .…”