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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| Published in: | Mathematical programming Vol. 200; no. 1; pp. 199 - 227 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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Springer Berlin Heidelberg
01.06.2023
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| ISSN: | 0025-5610, 1436-4646 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. |
| Audience | Academic |
| Author | Knop, Dušan Koutecký, Martin Onn, Shmuel Mnich, Matthias Levin, Asaf |
| Author_xml | – sequence: 1 givenname: Dušan surname: Knop fullname: Knop, Dušan organization: Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague – sequence: 2 givenname: Martin orcidid: 0000-0002-7846-0053 surname: Koutecký fullname: Koutecký, Martin organization: Computer Science Institute, Faculty of Mathematics and Physics, Charles University – sequence: 3 givenname: Asaf surname: Levin fullname: Levin, Asaf organization: Technion – Israel Institute of Technology – sequence: 4 givenname: Matthias surname: Mnich fullname: Mnich, Matthias email: matthias.mnich@tuhh.de organization: Hamburg University of Technology, Institute for Algorithms and Complexity – sequence: 5 givenname: Shmuel surname: Onn fullname: Onn, Shmuel organization: Technion – Israel Institute of Technology |
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| Cites_doi | 10.1287/moor.1110.0515 10.1007/BF02579456 10.1287/opre.28.6.1347 10.1007/s10107-014-0761-5 10.1002/(SICI)1099-1425(199806)1:1<55::AID-JOS2>3.0.CO;2-J 10.1007/BF01582130 10.1016/0377-2217(95)00022-I 10.1287/mnsc.18.1.98 10.1287/opre.39.4.648 10.1287/opre.9.6.849 10.1007/s10107-019-01402-2 10.1287/opre.47.6.862 10.1007/s10951-017-0550-0 10.1145/96559.96597 10.1145/3340322 10.1016/j.disopt.2014.07.003 10.1016/j.disopt.2006.06.006 10.1137/S1064827502401953 10.1016/j.jcss.2016.07.004 10.1112/blms/17.1.21 10.1145/7531.7535 10.1515/crll.1916.146.1 10.1287/ijoc.2017.0797 10.1145/3330137 10.1137/0213007 10.1137/S1052623497318682 10.1137/19M1303873 10.1016/0012-365X(94)00240-J 10.4171/093 10.1109/SFCS.1982.61 10.1137/1.9781611972443 10.1007/978-3-540-68279-0_13 10.1137/1.9781611976465.101 10.1007/978-3-642-13193-6_21 10.1007/978-3-642-78240-4 10.1007/s10107-013-0638-z 10.1007/978-3-642-38171-3_31 |
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| Keywords | Integer programming 90C10 Fixed-parameter algorithms Scheduling 49M27 90C27 Configuration IP |
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-fold integer programs (IPs) form an important class of block-structured IPs for which increasingly fast algorithms have recently been developed and... N-fold integer programs (IPs) form an important class of block-structured IPs for which increasingly fast algorithms have recently been developed and... |
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| SubjectTerms | 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 |
| Title | High-multiplicity N-fold IP via configuration LP |
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