Handbook of discrete and computational geometry
Certifying and repairing solutions to large LPs how good are LP-solvers?
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Computational Experience and the Explanatory Value of Condition Measures for Linear Optimization
SIAM Journal on Optimization
The branchwidth of graphs and their cycle matroids
Journal of Combinatorial Theory Series B
An algorithm to solve integer linear systems exactly using numerical methods
Journal of Symbolic Computation
Solving Very Sparse Rational Systems of Equations
ACM Transactions on Mathematical Software (TOMS)
Numeric-symbolic exact rational linear system solver
Proceedings of the 36th international symposium on Symbolic and algebraic computation
An exact rational mixed-integer programming solver
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Safe lower bounds for graph coloring
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Exact solutions to linear programming problems
Operations Research Letters
A branch-and-cut approach to the crossing number problem
Discrete Optimization
Operations Research Letters
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We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification of the objective function, right-hand side, and variable bounds. Exact computation is used to compute and store the exact representation of the transformed problems, while numeric computation is used for solving LPs. At all steps of the algorithm the LP bases encountered in the transformed problems correspond directly to LP bases in the original problem description. We demonstrate that this algorithm is effective in practice for computing extended precision solutions and that this leads to direct improvement of the best known methods for solving LPs exactly over the rational numbers.