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Complexity Issues on PTAS

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Handbook of Combinatorial Optimization

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Abstract

This chapter surveys various complexity issues of polynomial time approximation schemes (PTAS). Also under consideration are the refined subclasses of PTAS, including fully polynomial time approximation schemes (FPTAS) and efficient polynomial time approximation schemes (EPTAS). The key questions under consideration are as follows: what optimization problems have PTASs, FPTASs, and EPTASs, and how efficient are these algorithms?

On the positive side, there are syntactic classes for characterizing PTAS, FPTAS, and EPTAS. Expressing a problem as a member in these classes automatically gives an approximation scheme for the problem. For many graph problems on planar graphs, there are powerful graph-theory techniques for designing PTAS and EPTAS algorithms, which can be extended to some generalizations of planar graphs. On the negative side, there are convincing evidences that, for a large set of NP-hard optimization problems, efficient approximation schemes do not exist. Even for many problems that have PTASs or EPTASs, their running time has a strong lower bound.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Texas A&M University, 77843, College Station, TX, USA

    Jianer Chen

  2. Department of Computer Science, Lafayette College, 512 Acopian Engineering Center, 18042, Easton, PA, USA

    Ge Xia

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  1. Jianer Chen
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  2. Ge Xia
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Correspondence to Jianer Chen or Ge Xia .

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Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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Chen, J., Xia, G. (2013). Complexity Issues on PTAS. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_19

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