Simple and Fast Algorithms for Linear and Integer Programs With Two Variables per Inequality

@article{Hochbaum1994SimpleAF,
  title={Simple and Fast Algorithms for Linear and Integer Programs With Two Variables per Inequality},
  author={Dorit S. Hochbaum and Joseph Naor},
  journal={SIAM J. Comput.},
  year={1994},
  volume={23},
  pages={1179-1192}
}
The authors present an $O(mn^2 \log m)$ algorithm for solving feasibility in linear programs with up to two variables per inequality which is derived directly from the Fourier--Motzkin elimination method. [] Key Result However, it is shown that both a feasible solution and an optimal solution with respect to an arbitrary objective function can be computed in pseudo-polynomial time.

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References

SHOWING 1-10 OF 22 REFERENCES
Combinatorial algorithms for optimization problems
  • E. Cohen
  • Mathematics, Computer Science
  • 1992
TLDR
This thesis considers several combinatorial optimization problems that can be viewed as classes of linear programming problems with special structure and develops a technique that extends the classes of problems known to have strongly polynomial algorithms, or known to be quickly solvable in parallel.
An $n^{log n}$ algorithm for the two-variable-per-constraint linear programming satisfiability problem
TLDR
A simple algorithm is described which determines the satisfiability over the reals of a conjunction of linear inequalities, none of which contains more than two variables, which is particularly suited to applications in mechanical program verification.
A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality
  • Bengt Aspvall, Y. Shiloach
  • Mathematics, Computer Science
    20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
  • 1979
TLDR
A constructive algorithm for solving systems of linear inequalities (LI) with at most two variables per inequality with polynomial in the size of the input is presented.
Strongly polynomial-time and NC algorithms for detecting cycles in periodic graphs
TLDR
It is shown that strongly polynomial algorithms exist for any fixed dimension d and these algorithms also establish membership in the class NC.
The computational complexity of simultaneous Diophantine approximation problems
  • J. Lagarias
  • Mathematics, Computer Science
    23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
  • 1982
TLDR
It is shown that the problem of deciding whether a given vector α of rational numbers has a simultaneous approximation of specified accuracy with respect to the sup norm with denominator Q in a given interval 1 ≤ Q ≤ N is NP-complete.
A new approach to the maximum flow problem
TLDR
By incorporating the dynamic tree data structure of Sleator and Tarjan, a version of the algorithm running in O(nm log(n'/m)) time on an n-vertex, m-edge graph is obtained, as fast as any known method for any graph density and faster on graphs of moderate density.
Deciding Linear Inequalities by Computing Loop Residues
V R Pratt has shown that the real and integer feastbdlty of sets of linear mequallUes of the form x _< y + c can be decided quickly by examining the loops m certain graphs Pratt's method is
Maximal Closure of a Graph and Applications to Combinatorial Problems
This paper generalizes the selection problem discussed by J. M. Rhys [Rhys, J. M. W. 1970. Shared fixed cost and network flows. Management Sci.17 3, November.], J. D. Murchland [Murchland, J. D.
The Lattice Structure of Flow in Planar Graphs
TLDR
It is shown how to compactly encode the entire lattice and it is shown that the set of solutions to the min-cost flow problem forms a sublattice in the presented lattice.
Integer Programming with a Fixed Number of Variables
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.
...
...