Theory and Applications of n-Fold Integer Programming

Onn, Shmuel

doi:10.1007/978-1-4614-1927-3_20

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…For a concise introduction to Graver bases (and to the results on N -fold IPs), including short proofs to the main results, we refer the reader to the survey paper by Onn [12]. In this section, we state and prove results on Graver bases needed for the proof of our main theorem in the next section.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

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

Hemmecke¹,

Köppe

Weismantel

2010

Integer Programming and Combinatorial Optimization

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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.

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Section: Proof Of Main Resultsmentioning

confidence: 99%

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

Hemmecke¹,

Köppe

Weismantel

2010

Integer Programming and Combinatorial Optimization

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“…They are based on polynomial-size and -time reductions to the maximization of a (special) convex function over a system of constraints with an n-fold constraint matrix. The latter can be performed efficiently, which comes from a combination of recent algebraic Graver bases methods [12,13,18,19] and geometric edge-directions and zonotope methods [17,20]. We then transfer the complexity results to a generalized model for land consolidation, and present a new algorithm for it, in Section 4.…”

Section: Our Contributionsmentioning

confidence: 99%

“…As a service to the reader, let us briefly explain the main ingredients of this approach. For a more detailed background see the survey [19] and the books [12,18], as well as the papers [13,14,15]. The basic notion underlying these methods is the so-called Graver basis G( Ã(n) ) of the matrix Ã(n) .…”

Section: Algorithm and Efficiencymentioning

confidence: 99%

Efficient solutions for weight-balanced partitioning problems

Borgwardt

Onn

2016

Discrete Optimization

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We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to different features and the clusters adhere to given lower and upper bounds on the total weight of their points with respect to each of these features. Further the weight-contribution of a vector to a cluster can depend on the cluster it is assigned to. Our interest in these types of clustering problems is motivated by an application in land consolidation where the ability to perform this kind of balancing is crucial.Our framework maximizes an objective function that is convex in the summed-up utility of the items in each cluster. Despite hardness of convex maximization and many related problems, for fixed dimension and number of clusters, we are able to show that our clustering model is solvable in time polynomial in the number of items if the weightbalancing restrictions are defined using vectors from a fixed, finite domain. We conclude our discussion with a new, efficient model and algorithm for land consolidation.

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

2013

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We consider N -fold 4-block decomposable integer programs, which simultaneously generalize N -fold integer programs and two-stage stochastic integer programs with N scenarios. In previous work [R. Hemmecke, M. Köppe, R. Weismantel, A polynomial-time algorithm for optimizing over N -fold 4block decomposable integer programs, Proc. IPCO 2010, Lecture Notes in Computer Science, vol. 6080, Springer, 2010, it was proved that for fixed blocks but variable N , these integer programs are polynomial-time solvable for any linear objective. We extend this result to the minimization of separable convex objective functions. Our algorithm combines Graver basis techniques with a proximity result [D.S. Hochbaum and J.G. Shanthikumar, Convex separable optimization is not much harder than linear optimization, J. ACM 37 (1990), 843-862], which allows us to use convex continuous optimization as a subroutine.

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Theory and Applications of n-Fold Integer Programming

Cited by 4 publications

References 24 publications

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

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

Efficient solutions for weight-balanced partitioning problems

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

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