Synopsis: Under the widely-accepted impossibility of an efficient and exact method for an NP-hard optimization problem, a natural approach is to pursue an efficient but approximate method for a given NP-hard problem. Here, a key issue of this approximation approach is to guarantee a closeness of a solution to the optimum typically expressed in terms of an approximation factor. The goal of 2007-class is to survey the various design ideas of approximation algorithm. Students are also asked to apply these ideas to their own optimization problems.
An extended description: Towards essentially the same goal of a best possible approximation of a given problem, two mainstream approaches exist in approximation algorithm research: devising an algorithm of an improved approximation factor, and establishing the impossibility of an approximation with a certain approximation factor. The former has accumulated approximability results for the optimization problems arising in various fields of applications such as operations research, management science and computer science. The latter also has established many breakthrough results especially since 1990's on the basis of the probabilistic checkable proof.
Our 2007 Spring class, due to time constraint, will be focused on the former approach. Surveying the various design ideas of approximation algorithms, the students will be encouraged to explore an approximation algorithm of their own choice. In light of this goal, the Vazirani's book, Approximation Algorithms seems to offer an excellent exposition of wide range of approximation algorithms in a well balanced and organized manner. Besides the basic NP-completeness theories, we will cover a fair part of this book. Finally, if time permitted, we will review some applications of the gap-preserving reduction for inapproximability along with a sketch of the proof of the probabilistic checkable proof theorem.
Evaluation: 7 Homeworks (ABC grading), Final, Essay(optional).
(If an essay shows a certain level of originality and novelty, the student can get a good grade regardless of the homework and the final.)