CS 663: Pattern Matching Algorithms Scribe: Chen Jiang 11/29/2010 Summary: CS 663: Pattern Matching Algorithms Scribe: Chen Jiang 11/29/2010 PTAS for Bin-Packing 1. Introduction 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: )1()()( CIOPTCIA 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 Collections: Mathematics