Integer Programming with a Fixed Number of Variables

@article{Lenstra1983IntegerPW,
  title={Integer Programming with a Fixed Number of Variables},
  author={Hendrik W. Lenstra},
  journal={Math. Oper. Res.},
  year={1983},
  volume={8},
  pages={538-548}
}
  • H. Lenstra
  • Published 1 November 1983
  • Mathematics, Computer Science
  • Math. Oper. Res.
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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References

SHOWING 1-10 OF 23 REFERENCES
A Polynomial-Time Algorithm for the Knapsack Problem with Two Variables
TLDR
In this paper a very special knapsack problem is studied, namely, one with only two variables, and a polynomial-time algorithm is presented and analyzed.
Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix
Recently, Frumkin [9] pointed out that none of the well-known algorithms that transform an integer matrix into Smith [16] or Hermite [12] normal form is known to be polynomially bounded in its runn...
A Polynomial Algorithm for the Two-Variable Integer Programming Problem
  • R. Kannan
  • Mathematics, Computer Science
    JACM
  • 1980
ABSTRACT A polynomial time algorithm is presented for solving the following two-variable integer programming problem maximize ClXl + c2x2 subject to a, lxl + a,2x2 = O, integers, where a,j, cj, and
The ellipsoid method and its consequences in combinatorial optimization
TLDR
The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.
A bound on solutions of linear integer equalities and inequalities
Consider a system of linear equalities and inequalities with integer coefficients. We describe the set of rational solutions by a finite generating set of solution vectors. The entries of these
Factoring polynomials with rational coefficients
In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge
Production Sets with Indivisibilities Part I: Generalities
This paper and its sequel present a new approach to the study of production sets with indivisibilities and to the programming problems which arise when a factor endowment is specified. The absence of
An Introduction to the Geometry of Numbers
Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction
Systems of distinct representatives and linear algebra
So me purposes of thi s paper are: (1) To take se riously the term , " term rank. " (2) To ma ke an issue of not " rea rra nging rows a nd colu mns" by not "a rranging" the m in the firs t place. (3)
...
1
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3
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