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Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems

Summary: Polynomial Time Approximation Schemes for Dense
Instances of NP­Hard Problems
Sanjeev Arora # David Karger + Marek Karpinski #
We present a unified framework for designing polynomial time approximation schemes
(PTASs) for ``dense'' instances of many NP­hard optimization problems, including
maximum cut, graph bisection, graph separation, minimum k­way cut with and with­
out specified terminals, and maximum 3­satisfiability. By dense graphs we mean graphs
with minimum
degree## n), although our algorithms solve most of these problems so
long as the average degree is ## n). Denseness for non­graph problems is defined sim­
ilarly. 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.
1 Introduction
Approximation algorithms, whenever they can be found, are a way to deal with the NP­
hardness of optimization problems. Ideally, they should run in polynomial time and have a
small approximation ratio, which is the worst­case ratio of the value of the solution returned


Source: Arora, Sanjeev - Department of Computer Science, Princeton University


Collections: Computer Technologies and Information Sciences