 
Summary: The Approximability of NPhard Problems
Sanjeev Arora #
Princeton University
1 Introduction
Many problems in combinatorial optimization are
NPhard (see [60]). This has forced researchers to
explore techniques for dealing with NPcompleteness.
Some have considered algorithms that solve ``typi
cal'' or ``average'' instances instead of worstcase in
stances [86, 100]. In practice, however, identifying
``typical'' instances is not easy. Other researchers
have tried to design approximation algorithms. An
algorithm achieves an approximation ratio # for a
maximization problem if, for every instance, it pro
duces a solution of value at least OPT/#, where
OPT is the value of the optimal solution. (For a
minimization problem, achieving a ratio # involves
finding a solution of cost at most #OPT .) Note
that the approximation ratio is # 1 by definition.
After twentyfive years of research, approxima
