Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems

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Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems (1995)

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by Sanjeev Arora , David Karger , Marek Karpinski

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189 - 35 self

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@INPROCEEDINGS{Arora95polynomialtime,
    author = {Sanjeev Arora and David Karger and Marek Karpinski},
    title = {Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems},
    booktitle = {},
    year = {1995},
    pages = {284--293}
}

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Abstract

We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimum k-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for non-graph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.

Keyphrases

dense instance polynomial time approximation scheme np-hard problem unified framework dense graph constant degree many np-hard optimization problem maximum cut exhaustive sampling average degree graph separation algorithm solve small random set optimum solution graph bisection minimum k-way cut minimum degree omega dense polynomial certain smooth integer program objective function non-graph problem

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