Integer Programming with a Fixed Number of Variables

H. W. Lenstra, Jr.
H. W. Lenstra, Jr.
Mathematisch Instituut, Unrversiteit Van Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands
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Mathematisch Instituut, Unrversiteit Van Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands

Published Online:1 Nov 1983https://doi.org/10.1287/moor.8.4.538

Abstract

It is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable. The proof depends on methods from geometry of numbers.

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- On Structural Parameterizations of the Edge Disjoint Paths Problem
  25 January 2021 | Algorithmica, Vol. 83, No. 6
- Revenue maximization in Stackelberg Pricing Games: beyond the combinatorial setting
  18 April 2020 | Mathematical Programming, Vol. 187, No. 1-2
- EASE: Energy‐efficient task scheduling for edge computing under uncertain runtime and unstable communication conditions
  31 July 2019 | Concurrency and Computation: Practice and Experience, Vol. 33, No. 7
- A note on the concrete hardness of the shortest independent vector in lattices
  Information Processing Letters, Vol. 167
- New algorithms for minimizing the weighted number of tardy jobs on a single machine
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- Improved bi-criteria approximation schemes for load balancing on unrelated machines with cost constraints
  Theoretical Computer Science, Vol. 858
- Trichotomy for the reconfiguration problem of integer linear systems
  Theoretical Computer Science, Vol. 856
- On the status sequences of trees
  Theoretical Computer Science, Vol. 856
- r -Simple k -Path and Related Problems Parameterized by k / r
  21 January 2021 | ACM Transactions on Algorithms, Vol. 17, No. 1
- Minimizing maximum job dependent ergonomic risk
  27 November 2019 | International Journal of Production Research, Vol. 59, No. 1
- Meta-heuristic approaches to solve shortest lattice vector problem
  17 November 2019 | Journal of Discrete Mathematical Sciences and Cryptography, Vol. 24, No. 1
- Piecewise Linear Valued Constraint Satisfaction Problems with Fixed Number of Variables
  9 November 2020
- Parameterized Complexity of Defensive and Offensive Alliances in Graphs
  12 December 2020
- The Balanced Satisfactory Partition Problem
  11 January 2021
- Parameterized Complexity of Locally Minimal Defensive Alliances
  28 January 2021
- Acyclic Coloring Parameterized by Directed Clique-Width
  28 January 2021
- Computing L(p, 1)-Labeling with Combined Parameters
  16 February 2021
- On the Integrality Gap of Binary Integer Programs with Gaussian Data
  5 May 2021
- Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity
  4 May 2021
- Optimization for Project Scheduling
  13 December 2021
- Continuous Reformulation of Binary Variables, Revisited
  21 September 2021
- Can Romeo and Juliet Meet? or Rendezvous Games with Adversaries on Graphs
  20 September 2021
- EPTAS for Load Balancing Problem on Parallel Machines with a Non-renewable Resource
  1 January 2022
- On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem
  19 August 2020 | Algorithmica, Vol. 83, No. 1
- Multi-Omics Analysis in a Network Context
- Robustness among multiwinner voting rules
  Artificial Intelligence, Vol. 290
- An event-based reactive scheduling approach for the Resource Constrained Project Scheduling Problem with unreliable resources
  Computers & Industrial Engineering, Vol. 151
- Lattice Reformulation Cuts
  25 October 2021 | SIAM Journal on Optimization, Vol. 31, No. 4
- Polynomiality for Bin Packing with a Constant Number of Item Types
  7 November 2020 | Journal of the ACM, Vol. 67, No. 6
- Subgraph Isomorphism on Graph Classes that Exclude a Substructure
  2 July 2020 | Algorithmica, Vol. 82, No. 12
- Solving hard stable matching problems involving groups of similar agents
  Theoretical Computer Science, Vol. 844
- Closing the Gap for Makespan Scheduling via Sparsification Techniques
  Klaus Jansen,
  Kim-Manuel Klein,
  José Verschae
  16 June 2020 | Mathematics of Operations Research, Vol. 45, No. 4
- About the Structure of the Integer Cone and Its Application to Bin Packing
  Klaus Jansen,
  Kim-Manuel Klein
  14 July 2020 | Mathematics of Operations Research, Vol. 45, No. 4
- Combinatorial n-fold integer programming and applications
  20 November 2019 | Mathematical Programming, Vol. 184, No. 1-2
- A note on a variant of the online open end bin packing problem
  Operations Research Letters, Vol. 48, No. 6
- On degree sequence optimization
  Operations Research Letters, Vol. 48, No. 6
- Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions
  10 November 2020 | Mathematics, Vol. 8, No. 11
- References
  19 September 2020
- SRVB cryptosystem: another attempt to revive Knapsack-based public-key encryption schemes
- Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints
  1 June 2020 | ACM Transactions on Computation Theory, Vol. 12, No. 3
- Network control principles for identifying personalized driver genes in cancer
  10 November 2019 | Briefings in Bioinformatics, Vol. 21, No. 5
- Parameterized Multi-Scenario Single-Machine Scheduling Problems
  28 March 2020 | Algorithmica, Vol. 82, No. 9
- Adaptive streaming of HD and 360 ° videos over software defined radios
  Pervasive and Mobile Computing, Vol. 67
- Voting and Bribing in Single-Exponential Time
  22 June 2020 | ACM Transactions on Economics and Computation, Vol. 8, No. 3
- Time- and Space-optimal Algorithm for the Many-visits TSP
  6 June 2020 | ACM Transactions on Algorithms, Vol. 16, No. 3
- Electric vehicles in a smart grid: a comprehensive survey on optimal location of charging station
  22 April 2020 | IET Smart Grid, Vol. 3, No. 3
- Consensus Strings with Small Maximum Distance and Small Distance Sum
  4 November 2019 | Algorithmica, Vol. 82, No. 5
- Lattice closures of polyhedra
  19 March 2019 | Mathematical Programming, Vol. 181, No. 1
- An Evolutionary Approach to Constructing the Minimum Volume Ellipsoid Containing a Set of Points and the Maximum Volume Ellipsoid Embedded in a Set of Points
  Journal of Physics: Conference Series, Vol. 1530, No. 1
- Mixed integer programming with convex/concave constraints: Fixed-parameter tractability and applications to multicovering and voting
  Theoretical Computer Science, Vol. 814
- Optimal Perimeter Guarding With Heterogeneous Robot Teams: Complexity Analysis and Effective Algorithms
  IEEE Robotics and Automation Letters, Vol. 5, No. 2
- Exact and approximate algorithms for clustering problem in wireless sensor networks
  1 March 2020 | IET Communications, Vol. 14, No. 4
- F2MH Cryptosystem: Preliminary analysis of an original attempt to revive Knapsack-based public-key encryption schemes
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  9 March 2020 | ACM Transactions on Economics and Computation, Vol. 8, No. 1
- The Computational Complexity of Integer Programming with Alternations
  Danny Nguyen,
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  7 August 2019 | Mathematics of Operations Research, Vol. 45, No. 1
- A polynomial-time approximation scheme for an arbitrary number of parallel two-stage flow-shops
  European Journal of Operational Research, Vol. 281, No. 1
- Proximity Results and Faster Algorithms for Integer Programming Using the Steinitz Lemma
  15 November 2019 | ACM Transactions on Algorithms, Vol. 16, No. 1
- Parameterized Dynamic Variants of Red-Blue Dominating Set
  17 January 2020
- The Integrality Number of an Integer Program
  14 April 2020
- Iterated Type Partitions
  29 May 2020
- Slide Reduction, Revisited—Filling the Gaps in SVP Approximation
  10 August 2020
- Polyhedral Circuits and Their Applications
  9 August 2020
- Parameterized Complexity of Maximum Edge Colorable Subgraph
  27 August 2020
- Graph Square Roots of Small Distance from Degree One Graphs
  3 December 2020
- The Small Set Vertex Expansion Problem
  4 December 2020
- A Fine-Grained View on Stable Many-To-One Matching Problems with Lower and Upper Quotas
  6 December 2020
- On the Rational Polytopes with Chvátal Rank 1
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  Discrete Applied Mathematics, Vol. 272
- Minimization of Even Conic Functions on the Two-Dimensional Integral Lattice
  20 March 2020 | Journal of Applied and Industrial Mathematics, Vol. 14, No. 1
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- A novel network control model for identifying personalized driver genes in cancer
  25 November 2019 | PLOS Computational Biology, Vol. 15, No. 11
- Parameterized resiliency problems
  Theoretical Computer Science, Vol. 795
- An EPTAS for Scheduling on Unrelated Machines of Few Different Types
  10 May 2019 | Algorithmica, Vol. 81, No. 10
- Modified fuzzy c-mean for custom-sized clusters
  17 July 2019 | Sādhanā, Vol. 44, No. 8
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- High-multiplicity election problems
  4 May 2019 | Autonomous Agents and Multi-Agent Systems, Vol. 33, No. 4
- Knapsack problems: A parameterized point of view
  Theoretical Computer Science, Vol. 775
- Query Evaluation in Election Databases
  25 June 2019
- On approximating the covering radius and finding dense lattice subspaces
  23 June 2019
- An EPTAS for Machine Scheduling with Bag-Constraints
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  17 June 2019
- Virtual machine placement quality estimation in cloud infrastructures using integer linear programming
  25 October 2018 | Software Quality Journal, Vol. 27, No. 2
- The (not so) trivial lifting in two dimensions
  19 September 2018 | Mathematical Programming Computation, Vol. 11, No. 2
- Harmonious coloring: Parameterized algorithms and upper bounds
  Theoretical Computer Science, Vol. 772
- When Can Graph Hyperbolicity be Computed in Linear Time?
  13 October 2018 | Algorithmica, Vol. 81, No. 5
- Computing Weakly Reversible Deficiency Zero Network Translations Using Elementary Flux Modes
  21 February 2019 | Bulletin of Mathematical Biology, Vol. 81, No. 5
- Detecting Fault Injection Attacks Based on Compressed Sensing and Integer Linear Programming
  IEEE Transactions on Dependable and Secure Computing, Vol. 16, No. 3
- A Parameterized Algorithmics Framework for Degree Sequence Completion Problems in Directed Graphs
  21 August 2018 | Algorithmica, Vol. 81, No. 4
- On the complexity of quasiconvex integer minimization problem
  11 December 2018 | Journal of Global Optimization, Vol. 73, No. 4
- No-idle Parallel Machine Scheduling of Unit-time Jobs
- The Minimum Feasible Tileset Problem
  29 May 2018 | Algorithmica, Vol. 81, No. 3
- Scheduling two agents on a single machine: A parameterized analysis of NP-hard problems
  Omega, Vol. 83
- Estimation of the hardness of the learning with errors problem with a restricted number of samples
  8 August 2018 | Journal of Mathematical Cryptology, Vol. 13, No. 1
- Lamb Wave-Minimum Sampling Variance Particle Filter-Based Fatigue Crack Prognosis
  2 March 2019 | Sensors, Vol. 19, No. 5
- Tropical Principal Component Analysis and Its Application to Phylogenetics
  11 September 2018 | Bulletin of Mathematical Biology, Vol. 81, No. 2
- On a fully fuzzy framework for minimax mixed integer linear programming
  Computers & Industrial Engineering, Vol. 128
- A Fast Algorithm for Unbounded Monotone Integer Linear Systems with Two Variables per Inequality via Graph Decomposition
  21 December 2018
- Algorithms for Closest and Farthest String Problems via Rank Distance
  6 March 2019
- On Block-Structured Integer Programming and Its Applications
  1 June 2019
- Parameterized Complexity of Safe Set
  6 April 2019
- Subgraph Isomorphism on Graph Classes that Exclude a Substructure
  6 April 2019
- Integer Programming and Incidence Treedepth
  13 April 2019
- Integer Conic Function Minimization Based on the Comparison Oracle
  12 June 2019
- The Mersenne Low Hamming Combination Search Problem Can Be Reduced to an ILP Problem
  29 June 2019
- Parameterized Complexity of Asynchronous Border Minimization
  15 May 2018 | Algorithmica, Vol. 81, No. 1
- Minimum-Cost b-Edge Dominating Sets on Trees
  10 May 2018 | Algorithmica, Vol. 81, No. 1
- An Improved Attack on the Basic Merkle–Hellman Knapsack Cryptosystem
  IEEE Access, Vol. 7
- Reachability Analysis of Low‐Order Discrete State Reaction Networks Obeying Conservation Laws
  26 March 2019 | Complexity, Vol. 2019, No. 1
- FPT-Algorithm for Computing the Width of a Simplex Given by a Convex Hull
  2 May 2019 | Moscow University Computational Mathematics and Cybernetics, Vol. 43, No. 1
- Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities
  1 January 2019 | Econometrica, Vol. 87, No. 3
- On the number of integer points in a multidimensional domain
  Discrete Mathematics and Applications, Vol. 28, No. 6
- JALAD: Joint Accuracy-And Latency-Aware Deep Structure Decoupling for Edge-Cloud Execution
- Review of recent trends in charging infrastructure planning for electric vehicles
  6 June 2018 | WIREs Energy and Environment, Vol. 7, No. 6
- On approximation algorithms for concave mixed-integer quadratic programming
  25 July 2017 | Mathematical Programming, Vol. 172, No. 1-2
- Pisot unit generators in number fields
  Journal of Symbolic Computation, Vol. 89
- Toward a Smart Car: Hybrid Nonlinear Predictive Controller With Adaptive Horizon
  IEEE Transactions on Control Systems Technology, Vol. 26, No. 6
- Multiwinner analogues of the plurality rule: axiomatic and algorithmic perspectives
  19 April 2018 | Social Choice and Welfare, Vol. 51, No. 3
- Scheduling meets n-fold integer programming
  15 November 2017 | Journal of Scheduling, Vol. 21, No. 5
- Multi-attribute proportional representation
  Artificial Intelligence, Vol. 263
- Parameterized complexity of distance labeling and uniform channel assignment problems
  Discrete Applied Mathematics, Vol. 248
- How hard is it to control a group?
  11 June 2018 | Autonomous Agents and Multi-Agent Systems, Vol. 32, No. 5
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  5 October 2017 | Journal of Scheduling, Vol. 21, No. 4
- A survival guide to presburger arithmetic
  26 July 2018 | ACM SIGLOG News, Vol. 5, No. 3
- (Gap/S)ETH hardness of SVP
  20 June 2018
- Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix
  19 April 2017 | Mathematical Programming, Vol. 169, No. 2
- Swapping colored tokens on graphs
  Theoretical Computer Science, Vol. 729
- Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming
  Nathan Adelgren,
  Pietro Belotti,
  Akshay Gupte
  30 April 2018 | INFORMS Journal on Computing, Vol. 30, No. 2
- FPT-algorithms for some problems related to integer programming
  17 February 2018 | Journal of Combinatorial Optimization, Vol. 35, No. 4
- Active instance matching with pairwise constraints and its application to Chinese knowledge base construction
  30 June 2017 | Knowledge and Information Systems, Vol. 55, No. 1
- The complexity landscape of decompositional parameters for ILP
  Artificial Intelligence, Vol. 257
- Super-Golden-Gates for PU(2)
  Advances in Mathematics, Vol. 327
- On the string consensus problem and the Manhattan sequence consensus problem
  Theoretical Computer Science, Vol. 710
- Underminer
  6 December 2017 | ACM Transactions on Embedded Computing Systems, Vol. 17, No. 1
- New (and Old) Proof Systems for Lattice Problems
  1 March 2018
- Learning with Errors and Extrapolated Dihedral Cosets
  1 March 2018
- Satisfying Neighbor Preferences on a Circle
  13 March 2018
- Two-Dimensional Knapsack for Circles
  13 March 2018
- A Unified Framework for Designing EPTAS’s for Load Balancing on Parallel Machines
  3 July 2018
- Using Structural Properties for Integer Programs
  3 July 2018
- Bin-Packing
  14 March 2018
- Bin-Packing
  25 July 2018
- Local Testing of Lattices
  SIAM Journal on Discrete Mathematics, Vol. 32, No. 2
- Enumerating Projections of Integer Points in Unbounded Polyhedra
  SIAM Journal on Discrete Mathematics, Vol. 32, No. 2
- Equivalence of Lattice Orbit Polytopes
  SIAM Journal on Applied Algebra and Geometry, Vol. 2, No. 2
- The Support of Integer Optimal Solutions
  SIAM Journal on Optimization, Vol. 28, No. 3
- Improved approximation algorithms for parallel machine scheduling with release dates and job rejection
  19 December 2016 | 4OR, Vol. 15, No. 4
- Deadlock and liveness characterization for a class of generalized Petri nets
  Information Sciences, Vol. 420
- Polynomial-time approximation for nonconvex optimization problems with an L1-constraint
- The duality principle in the theory of development generalized A-M knapsack cryptosystems
  13 October 2017
- On the Quantitative Hardness of CVP
- Short Presburger Arithmetic Is Hard
- Complexity and Approximability of Parameterized MAX-CSPs
  19 April 2017 | Algorithmica, Vol. 79, No. 1
- About the Decidability of Polyhedral Separability in the Lattice $$\mathbb {Z}^d$$ Z d
  7 March 2017 | Journal of Mathematical Imaging and Vision, Vol. 59, No. 1
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  26 September 2017 | Journal of Statistical Mechanics: Theory and Experiment, Vol. 2017, No. 9
- Processing Succinct Matrices and Vectors
  26 December 2015 | Theory of Computing Systems, Vol. 61, No. 2
- Approximation Algorithms for Maximin Fair Division
  20 June 2017
- Complexity of short Presburger arithmetic
  19 June 2017
- Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems
  26 April 2017 | Annals of Combinatorics, Vol. 21, No. 2
- Alliances in graphs of bounded clique-width
  Discrete Applied Mathematics, Vol. 223
- Autogen
  5 October 2017 | ACM Transactions on Parallel Computing, Vol. 4, No. 1
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  Journal of Computer and System Sciences, Vol. 84
- A Review of Modern Cryptography: From the World War II Era to the Big-Data Era
  6 May 2017
- Partitioning Graphs into Induced Subgraphs
  16 February 2017
- Parameterized Resiliency Problems via Integer Linear Programming
  14 April 2017
- New Algorithmic Results for Bin Packing and Scheduling
  14 April 2017
- Enumeration of Integer Points in Projections of Unbounded Polyhedra
  24 May 2017
- An EPTAS for Scheduling on Unrelated Machines of Few Different Types
  5 July 2017
- Robustness Among Multiwinner Voting Rules
  19 August 2017
- Thinner is not always better: Cascade knapsack problems
  Operations Research Letters, Vol. 45, No. 1
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  SIAM Journal on Optimization, Vol. 27, No. 2
- Minimum dimensional Hamming embeddings
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- О количестве целых точек в многомерной области
  5 December 2017 | Дискретная математика, Vol. 29, No. 4
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  19 January 2016 | Algorithmica, Vol. 76, No. 4
- Prices matter for the parameterized complexity of shift bribery
  Information and Computation, Vol. 251
- Integer programming as projection
  Discrete Optimization, Vol. 22
- Polynomial-Time Approximation Schemes for Circle and Other Packing Problems
  21 August 2015 | Algorithmica, Vol. 76, No. 2
- High-multiplicity scheduling on one machine with forbidden start and completion times
  17 March 2016 | Journal of Scheduling, Vol. 19, No. 5
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  Operations Research Letters, Vol. 44, No. 5
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  IEEE Transactions on Power Systems, Vol. 31, No. 5
- Robust Polynomial-Time Approximation Schemes for Parallel Machine Scheduling with Job Arrivals and Departures
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  José Verschae
  11 April 2016 | Mathematics of Operations Research, Vol. 41, No. 3
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  3 November 2015 | Annals of Mathematics and Artificial Intelligence, Vol. 77, No. 3-4
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  Control Engineering Practice, Vol. 53
- On polynomial kernels for sparse integer linear programs
  Journal of Computer and System Sciences, Vol. 82, No. 5
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  5 May 2016 | ACM Transactions on Computation Theory, Vol. 8, No. 2
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  Robert Hildebrand,
  Robert Weismantel,
  Kevin Zemmer
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  2 March 2016 | Communications in Algebra, Vol. 44, No. 4
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  Theoretical Computer Science, Vol. 622
- A Dichotomy for Real Weighted Holant Problems
  17 November 2015 | computational complexity, Vol. 25, No. 1
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  5 January 2016 | ACM Transactions on Economics and Computation, Vol. 4, No. 1
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  22 April 2016
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  22 April 2016
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  22 April 2016
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  9 April 2016
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  25 May 2016
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  25 May 2016
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  25 May 2016
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  31 May 2016
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  5 July 2016
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  20 July 2016
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  28 September 2016
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  Sanjay Mehrotra
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  6 October 2015 | Fixed Point Theory and Applications, Vol. 2015, No. 1
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  Theoretical Computer Science, Vol. 607
- Resampling methods for particle filtering: identical distribution, a new method, and comparable study
  26 October 2015 | Frontiers of Information Technology & Electronic Engineering, Vol. 16, No. 11
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  Theoretical Computer Science, Vol. 600
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  IEEE Transactions on Automation Science and Engineering, Vol. 12, No. 3
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  15 June 2015
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  14 June 2015
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  Journal of Computer and System Sciences, Vol. 81, No. 4
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  Simone Falavigna,
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  Operations Research Letters, Vol. 43, No. 3
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  27 August 2014
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  21 January 2015
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  16 April 2015
- On the Complexity of Wafer-to-Wafer Integration
  16 May 2015
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  23 April 2015
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  4 August 2015
- More Natural Models of Electoral Control by Partition
  28 August 2015
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  28 August 2015
- Arbitrage in Frictional Foreign Exchange Market
  30 May 2015
- Polynomial Kernels for Weighted Problems
  11 August 2015
- Algorithmic Applications of Tree-Cut Width
  11 August 2015
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  Operations Research Letters, Vol. 43, No. 1
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  SIAM Journal on Discrete Mathematics, Vol. 29, No. 4
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  SSRN Electronic Journal, Vol. 13
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  Discrete Applied Mathematics, Vol. 179
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  20 September 2013 | Algorithmica, Vol. 70, No. 2
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  Operations Research Letters, Vol. 42, No. 6-7
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  Frederik von Heymann
  14 November 2013 | Mathematics of Operations Research, Vol. 39, No. 3
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  13 August 2014 | Automation and Remote Control, Vol. 75, No. 8
- General Bibliography
  8 August 2014
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  8 August 2014
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  1 February 2014 | Journal of Global Optimization, Vol. 59, No. 2-3
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  Applied Mathematical Modelling, Vol. 38, No. 14
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  28 March 2013 | Mathematical Programming, Vol. 145, No. 1-2
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- From Angular Manifolds to the Integer Lattice: Guaranteed Orientation Estimation With Application to Pose Graph Optimization
  IEEE Transactions on Robotics, Vol. 30, No. 2
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  6 April 2013 | Optimization Letters, Vol. 8, No. 3
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  Discrete Applied Mathematics, Vol. 165
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  Discrete Mathematics, Vol. 315-316
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  26 September 2014
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  26 September 2014
- Linear Inequalities and Polyhedra
  26 September 2014
- Valid Inequalities for Structured Integer Programs
  26 September 2014
- Reformulations and Relaxations
  26 September 2014
- Enumeration
  26 September 2014
- On the String Consensus Problem and the Manhattan Sequence Consensus Problem
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  8 November 2014
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  17 October 2014
- Efficient Polynomial Time Approximation Scheme for Scheduling Jobs on Uniform Processors
  17 October 2014
- Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds
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cover image Mathematics of Operations Research

Volume 8, Issue 4

November 1983

Pages 479-636

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Published Online:November 01, 1983

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H. W. Lenstra, Jr., (1983) Integer Programming with a Fixed Number of Variables. Mathematics of Operations Research 8(4):538-548.

https://doi.org/10.1287/moor.8.4.538

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Integer Programming with a Fixed Number of Variables

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Volume 8, Issue 4

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