Abstract
We consider a generic paradigm to design improved PTAS for linear programming relaxations of combinatorial optimization problems. For the case of the uncapacitated facility location problem, the scheduling problem R ∥ C max and the set covering problem we substantially improve the running time dependence on ε from the previously known O(1/ε 2) to O(1/ε). All these algorithms are remarkably simple to implement. For the survivable network design problem we improve the dependence from O(1/ε 2) to O((1/ε) log (1/ε)). We present some preliminary computational results that seem to suggest that our algorithms may prove very competitive in practice. Furthermore, we present a general approximation version of a result of Nesterov that potentially can be applied to a very large class of linear programming problems.
Our results build mainly on work of Nesterov and extend the work of Bienstock and Iyengar. In fact, one of the objectives of this paper is to make clearer the relationship between these two.
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Chudak, F.A., Eleutério, V. (2005). Improved Approximation Schemes for Linear Programming Relaxations of Combinatorial Optimization Problems. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_7
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DOI: https://doi.org/10.1007/11496915_7
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