 
Summary: Polynomial Time Approximation Schemes for Dense
Instances of NPHard Problems
Sanjeev Arora # David Karger + Marek Karpinski #
Abstract
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
