Polynomial Time Approximation Schemes for Dense Instances of $\text{N}$$\text{P}$-Hard Problems

Abstract

We present a unified framework for designing polynomial time approximation schemes (PTASs) for “dense” instances of many $\text{N}$$\text{P}$-hard optimization problems, including maximum cut, graph bisection, graph separation, minimumk-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degreeΩ(n), although our algorithms solve most of these problems so long as the average degree isΩ(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea ofexhaustive 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 certainsmoothinteger programs, where the objective function and the constraints are “dense” polynomials of constant degree.