The Design of Approximation Algorithms by Williamson & Shmoys (http://www.designofapproxalgs.com/) is a great book for many approximation methods such as greedy algorithms, semidefinite programming, etc. Also, it covers some topics within complexity that are closely-related to approximation algorithms (inapproximability, Unique Games-based hardness of MAX-CUT).

The Design of Approximation Algorithms by Williamson & Shmoys (http://www.designofapproxalgs.com/) is a great book for many approximation methods such as greedy algorithms, semidefinite programming, etc. Also, it covers some topics within complexity that are closely related to approximation algorithms (inapproximability, Unique Games-based hardness of MAX-CUT).