Summary: CS 663: Pattern Matching Algorithms
Scribe: Chen Jiang 11/29/2010
PTAS for Bin-Packing
The Bin-Packing problem is NP-hard. If we use approximation algorithms, the Bin-Packing
problem could be solved in polynomial time. For example, the simplest approximation algorithm
is the First-fit algorithm, which solves the Bin-Packing problem in time )log( nnO .
We use the approximation factor to determine how good our approximation algorithm is.
Let )(IA be the number of bins required by the approximation algorithm, and let )(IOPT be
the optimal number of required bins for input I. We say that algorithm A has approximation factor
C if for every input I:
This inequality means that the approximation algorithm would not use more than C times the
optimal number of bins, which is also the upper bound of the approximation algorithm.
Obviously, the closer C is to 1, the better the approximation.
Claim: The Bin-Packing problem has a PTAS (Polynomial Time Approximation Scheme).
I.e., Given 0> , one can always produce approximation algorithm whose (1) time is
polynomial in n and ; and (2) whose approximation factor is,
)()1()( IOPTIA +
2. Special Cases for Bin-Packing