
University of Bonn > Department of Computer Science > Chair V  
CSAPXReports 1998  Copyright 1998 University of Bonn, Department of Computer Science, Abt. V  
8950

Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
Sanjeev Arora, David Karger, Marek Karpinski [Download PostScript] [Download PDF] 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 3satisfiability. By dense graphs we mean graphs with minimum degree $\Omega(n)$, although %\editsmall{some of our algorithms work}{our algorithms solve most of these problems so long as the average degree is $\Omega(n)$. Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of {\em 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 {\em smooth\/} integer programs where the objective function and the constraints are `dense'' polynomials of constant degree. 

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University of Bonn > Department of Computer Science > Chair V 