Please describe a complex problem in C++ that involves the use of object-oriented programming principles, algorithms, and data structures. Also, provide a step-by-step solution to the problem.

Problem: Develop a program that simulates a library management system. The program should allow librarians to add, delete, and modify books in the library's inventory, keep track of which books are checked out, and generate reports of the most popular books and the books that are overdue. The library's inventory should be stored in a database, and the program should be able to search the database to retrieve information about specific books