Scheduling Kernels via Configuration LP

Knop, Dušan; Koutecký, Martin

doi:10.48550/arxiv.2003.02187

Cited by 3 publications

(3 citation statements)

References 28 publications

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“…IPs of this kind appear in various contexts, see e.g. [26,34,35,40]. These (theoretical) tractability results complement well a vast number of empirical results demonstrating tractability of instances with a block structure, e.g.…”

Section: Introductionsupporting

confidence: 63%

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Briański¹,

Koutecký²,

Kráľ³

et al. 2022

Preprint

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An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively.We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ 1 -norm of the Graver basis is bounded by a function of the maximum ℓ 1 -norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists.Our results yield parameterized algorithms for integer programming when parameterized by the ℓ 1 -norm of the Graver basis of the constraint matrix, when parameterized by the ℓ 1 -norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix.

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Section: Introductionsupporting

confidence: 63%

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Briański¹,

Koutecký²,

Kráľ³

et al. 2022

Preprint

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show abstract

“…An extended version [31] of this paper shows how to model many scheduling problems as high multiplicity N -fold IPs, so that an application of Theorem 1 yields new parameterized algorithms for these problems. Knop and Koutecký [30] use our new proximity theorem to show efficient preprocessing algorithms (kernels) for scheduling problems.…”

Section: Related Workmentioning

confidence: 99%

High-multiplicity N-fold IP via configuration LP

et al. 2022

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N-fold integer programs (IPs) form an important class of block-structured IPs for which increasingly fast algorithms have recently been developed and successfully applied. We study high-multiplicityN-fold IPs, which encode IPs succinctly by presenting a description of each block type and a vector of block multiplicities. Our goal is to design algorithms which solve N-fold IPs in time polynomial in the size of the succinct encoding, which may be significantly smaller than the size of the explicit (non-succinct) instance. We present the first fixed-parameter algorithm for high-multiplicity N-fold IPs, which even works for convex objectives. Our key contribution is a novel proximity theorem which relates fractional and integer optima of the Configuration LP, a fundamental notion by Gilmore and Gomory [Oper. Res., 1961] which we generalize. Our algorithm for N-fold IP is faster than previous algorithms whenever the number of blocks is much larger than the number of block types, such as in N-fold IP models for various scheduling problems.

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“…Our results and outline of this work. Contributing to the still scarce body of work on data reduction for packing and scheduling [4,16,24,28], we study the potential for polynomial-time data reduction for DVBP.…”

Section: Introductionmentioning

confidence: 99%

On data reduction for dynamic vector bin packing

Bevern¹,

Melnikov²,

Smirnov³

et al. 2022

Preprint

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We study a dynamic vector bin packing (DVBP) problem. We show hardness for shrinking arbitrary DVBP instances to size polynomial in the number of request types or in the maximal number of requests overlapping in time. We also present a simple polynomial-time data reduction algorithm that allows to recover (1 + ε)-approximate solutions for arbitrary ε > 0. It shrinks instances from Microsoft Azure and Huawei Cloud by an order of magnitude for ε = 0.02.

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Scheduling Kernels via Configuration LP

Cited by 3 publications

References 28 publications

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

High-multiplicity N-fold IP via configuration LP

On data reduction for dynamic vector bin packing

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