research-article

Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma

Authors:

Friedrich Eisenbrand,

Robert WeismantelAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 16, Issue 1

Article No.: 5, Pages 1 - 14

https://doi.org/10.1145/3340322

Published: 15 November 2019 Publication History

Abstract

We consider integer programming problems in standard form max { c^Tx : Ax = b, x⩾ 0, x ∈ Zⁿ} where A ∈ Z^{m × n}, b ∈ Z^m, and c ∈ Zⁿ. We show that such an integer program can be solved in time (m ⋅ Δ)^O(m) ⋅ \Vert b\Vert_∞², where Δ is an upper bound on each absolute value of an entry in A. This improves upon the longstanding best bound of Papadimitriou [27] of (m⋅ Δ)^O(m²), where in addition, the absolute values of the entries of b also need to be bounded by Δ. Our result relies on a lemma of Steinitz that states that a set of vectors in R^m that is contained in the unit ball of a norm and that sum up to zero can be ordered such that all partial sums are of norm bounded by m.

We also use the Steinitz lemma to show that the ℓ₁-distance of an optimal integer and fractional solution, also under the presence of upper bounds on the variables, is bounded by m ⋅ (2,m⋅ Δ +1)^m. Here Δ is again an upper bound on the absolute values of the entries of A. The novel strength of our bound is that it is independent of n.

We provide evidence for the significance of our bound by applying it to general knapsack problems where we obtain structural and algorithmic results that improve upon the recent literature.

References

[1]

A. V. Aho, J. E. Hopcroft, and J. D. Ullman. 1974. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reaing.

[2]

I. Aliev, M. Henk, and T. Oertel. 2017. Integrality gaps of integer knapsack problems. In International Conference on Integer Programming and Combinatorial Optimization. Springer, 25--38.

[3]

W. Banaszczyk. 1998. Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Structures and Algorithms 12, 4 (1998), 351--360.

Digital Library

[4]

W. Banaszczyk. 2012. On series of signed vectors and their rearrangements. Random Structures and Algorithms 40, 3 (2012), 301--316.

Digital Library

[5]

N. Bansal, D. Dadush, and S. Garg. 2016. An algorithm for Komlós conjecture matching Banaszczyk’s bound. In IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS’16). IEEE, 788--799.

[6]

N. Bansal and S. Garg. 2017. Algorithmic discrepancy beyond partial coloring. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. ACM, 914--926.

[7]

I. Bárány. 2008. On the power of linear dependencies. In Building Bridges. Springer, 31--45.

[8]

J. Beck and T. Fiala. 1981. “Integer-making” theorems. Discrete Applied Mathematics 3, 1 (1981), 1--8.

[9]

I. Borosh and L. B. Treybig. 1976. Bounds on positive integral solutions of linear diophantine equations. Proceedings of the American Mathematical Society 55, 2 (1976), 299--304.

[10]

K. Buchin, J. Matoušek, R. A. Moser, and D. Pálvölgyi. 2012. Vectors in a box. Mathematical Programming 135, 1--2 (2012), 323--335.

Digital Library

[11]

W. Cook, A. M. H. Gerards, A. Schrijver, and E. Tardos. 1986. Sensitivity theorems in integer linear programming. Mathematical Programming 34 (1986), 251--264.

Digital Library

[12]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. 2001. Introduction to Algorithms (2nd ed.). MIT Press, Cambridge, MA.

[13]

D. N. Dadush. 2012. Integer Programming, Lattice Algorithms, and Deterministic Volume Estimation. Georgia Institute of Technology.

[14]

S. Dash, R. Fukasawa, and O. Günlük. 2012. The master equality polyhedron with multiple rows. Mathematical Programming 132, 1--2 (2012), 125--151.

Digital Library

[15]

F. V. Fomin, F. Panolan, M. Ramanujan, and S. Saurabh. 2016. Fine-grained complexity of integer programming: The case of bounded branch-width and rank. arXiv preprint arXiv:1607.05342 (2016).

[16]

V. S. Grinberg and S. V. Sevast’yanov. 1980. Value of the Steinitz constant. Functional Analysis and Its Applications 14, 2 (1980), 125--126.

[17]

D. S. Hochbaum and D. B. Shmoys. 1987. Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the ACM 34,1 (Jan. 1987), 144--162.

Digital Library

[18]

O. H. Ibarra and C. E. Kim. 1975. Fast approximation algorithms for the knapsack and sum of subset problems. Journal of the Association for Computing Machinery 22, 4 (1975), 463--468.

Digital Library

[19]

K. Jansen. 2010. An EPTAS for scheduling jobs on uniform processors: Using an MILP relaxation with a constant number of integral variables. SIAM Journal on Discrete Mathematics 24, 2 (2010), 457--485.

Digital Library

[20]

K. Jansen, K.-M. Klein, and J. Verschae. 2016. Closing the gap for makespan scheduling via sparsification techniques. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP ’16), volume 55 of LIPIcs. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

[21]

K. Jansen and L. Rohwedder. 2018. On integer programming and convolution. In 10th Innovations in Theoretical Computer Science Conference (ITCS’19). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

[22]

R. Kannan. 1987. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research 12, 3 (1987), 415--440.

Digital Library

[23]

D. Knop, M. Pilipczuk, and M. Wrochna. 2018. Tight complexity lower bounds for integer linear programming with few constraints. arXiv preprint arXiv:1811.01296 (2018).

[24]

H. W. Lenstra. 1983. Integer programming with a fixed number of variables. Mathematics of Operations Research 8, 4 (1983), 538--548.

Digital Library

[25]

M. Lewenstein, S. Pettie, and V. Vassilevska Williams. 2017. Structure and hardness in p (Dagstuhl seminar 16451). In Dagstuhl Reports, volume 6:11. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

[26]

J. Matousek. 2009. Geometric Discrepancy: An Illustrated Guide, Vol. 18. Springer Science 8 Business Media.

[27]

C. H. Papadimitriou. 1981. On the complexity of integer programming. Journal of the Association for Computing Machinery 28, 4 (1981), 765--768.

Digital Library

[28]

U. Pferschy. 1999. Dynamic programming revisited: Improving knapsack algorithms. Computing 63, 4 (1999), 419--430.

Digital Library

[29]

P. Sanders, N. Sivadasan, and M. Skutella. 2009. Online scheduling with bounded migration. Mathematics of Operations Research 34, 2 (2009), 481--498.

Digital Library

[30]

A. Schrijver. 1986. Theory of Linear and Integer Programming. John Wiley.

[31]

S. Sevast’janov. 1978. Approximate solution of some problems of scheduling theory. Metody Diskretnogi Analiza 32 (1978), 66--75.

[32]

S. Sevast’janov and W. Banaszczyk. 1997. To the Steinitz lemma in coordinate form. Discrete Mathematics 169, 1--3 (1997), 145--152.

Digital Library

[33]

E. Steinitz. 1913. Bedingt konvergente Reihen und konvexe Systeme. Journal für die reine und angewandte Mathematik 143 (1913), 128--176.

[34]

A. Tamir. 2009. New pseudopolynomial complexity bounds for the bounded and other integer knapsack related problems. Operations Research Letters 37, 5 (2009), 303--306.

Digital Library

Cited By

Fiorini SJoret GWeltge SYuditsky Y(2025)Integer programs with bounded subdeterminants and two nonzeros per rowJournal of the ACM10.1145/369598572:1(1-50)Online publication date: 24-Jan-2025
https://doi.org/10.1145/3695985
Bach EEisenbrand FPinchasi R(2025)Integer points in the degree-sequence polytopeDiscrete Optimization10.1016/j.disopt.2024.10086755(100867)Online publication date: Feb-2025
https://doi.org/10.1016/j.disopt.2024.100867
Celaya MKuhlmann SPaat JWeismantel R(2024)Proximity and Flatness Bounds for Linear Integer OptimizationMathematics of Operations Research10.1287/moor.2022.033549:4(2446-2467)Online publication date: Nov-2024
https://doi.org/10.1287/moor.2022.0335
Show More Cited By

Index Terms

Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma
1. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Discrete optimization

Recommendations

Proximity results and faster algorithms for integer programming using the steinitz lemma
SODA '18: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

We consider integer programming problems in standard form max{c^T x : Ax = b, x ≥ 0, x ϵ Zⁿ} where A ϵ Z^m×n, b ϵ Z^m and c ϵ Zⁿ. We show that such an integer program can be solved in time [EQUATION], where Δ is an upper bound on each absolute value of an ...
Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem

In this paper, we consider problem (P) of minimizing a quadratic function q(x)=xtQx+ctx of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective ...
Cross Decomposition Applied to Integer Programming Problems: Duality Gaps and Convexification in Parts

<P>We study the lower bounds on the optimal objective function value of linear pure integer programming problems obtainable by the convexification in parts that results from using Benders' or cross decomposition, and the best lower bounds obtainable by ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 16, Issue 1

Special Issue on Soda'18 and Regular Papers

January 2020

369 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3372373

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2019 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 15 November 2019

Accepted: 01 June 2019

Revised: 01 June 2019

Received: 01 March 2018

Published in TALG Volume 16, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

SNSF

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

44
Total Citations
View Citations
609
Total Downloads

Downloads (Last 12 months)102
Downloads (Last 6 weeks)5

Reflects downloads up to 04 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Fiorini SJoret GWeltge SYuditsky Y(2025)Integer programs with bounded subdeterminants and two nonzeros per rowJournal of the ACM10.1145/369598572:1(1-50)Online publication date: 24-Jan-2025
https://doi.org/10.1145/3695985
Bach EEisenbrand FPinchasi R(2025)Integer points in the degree-sequence polytopeDiscrete Optimization10.1016/j.disopt.2024.10086755(100867)Online publication date: Feb-2025
https://doi.org/10.1016/j.disopt.2024.100867
Celaya MKuhlmann SPaat JWeismantel R(2024)Proximity and Flatness Bounds for Linear Integer OptimizationMathematics of Operations Research10.1287/moor.2022.033549:4(2446-2467)Online publication date: Nov-2024
https://doi.org/10.1287/moor.2022.0335
Hermelin DMolter HShabtay D(2024)Minimizing the Weighted Number of Tardy Jobs via (max,+)-ConvolutionsINFORMS Journal on Computing10.1287/ijoc.2022.030736:3(836-848)Online publication date: 1-May-2024
https://dl.acm.org/doi/10.1287/ijoc.2022.0307
Chen LLian JMao YZhang GMohar BShinkar IO'Donnell R(2024)A Nearly Quadratic-Time FPTAS for KnapsackProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649730(283-294)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649730
Bringmann KMohar BShinkar IO'Donnell R(2024)Knapsack with Small Items in Near-Quadratic TimeProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649719(259-270)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649719
Jin CMohar BShinkar IO'Donnell R(2024)0-1 Knapsack in Nearly Quadratic TimeProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649618(271-282)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649618
Eisenbrand FRohwedder LWęgrzycki K(2024)Sensitivity, Proximity and FPT Algorithms for Exact Matroid Problems2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00100(1610-1620)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00100
Chistikov DCzerwiński WMazowiecki FOrlikowski ŁSinclair-Banks HWęgrzycki K(2024)The Tractability Border of Reachability in Simple Vector Addition Systems with States2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00086(1332-1354)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00086
Jaykrishnan GLevin A(2024)Scheduling with cardinality dependent unavailability periodsEuropean Journal of Operational Research10.1016/j.ejor.2024.02.038316:2(443-458)Online publication date: Jul-2024
https://doi.org/10.1016/j.ejor.2024.02.038
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Figures

Tables

Media

View Issue’s Table of Contents