High-multiplicity N-fold IP via configuration LP

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-multiplicity N -fold IPs, which encode IPs succinctly by presenting a description of each block type and a vector...

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Veröffentlicht in:Mathematical programming Jg. 200; H. 1; S. 199 - 227
Hauptverfasser: Knop, Dušan, Koutecký, Martin, Levin, Asaf, Mnich, Matthias, Onn, Shmuel
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2023
Springer
Schlagworte:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Integer programming
90C10
Fixed-parameter algorithms
Scheduling
49M27
90C27
Configuration IP
ISSN:0025-5610, 1436-4646
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Zusammenfassung: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-multiplicity N -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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-022-01882-9