The airport in Patna, one of India’s smaller cities and the capital of its poorest state, is so crowded that government authorities this year said they need to build another one. The airport has seen the number of passengers, and daily flights, double in the last four years as the country’s largest carrier IndiGo Airlines and smaller rivals like GoAir seek to cash in on a surge in demand for air travel from smaller regions to offset losses on highly competitive “trunk” routes linking big cities.