What is the ruling on commencing the fast and breaking it in non-Islamic countries where there is no accredited Islamic center, and where some groups rely on astronomy while others follow the Kingdom of Saudi Arabia?

The determining factor for the sighting of the crescent moon for Ramadan and Shawwal is the Sharia-compliant sighting, not astronomical calculations. The اختلاف المطالع (variation in moonrise or horizons) should be taken into consideration. Muslims must strive for unity and agreement. If there is no Islamic center in their country, they should form a Sharia committee or follow the nearest neighboring country in sighting the crescent moon. One who is far from a specific country is not obliged to follow it in fasting or breaking the fast. The disagreement regarding the issue of اختلاف المطالع is an old and valid one, and it is not blameworthy if it is based on ijtihad (independent reasoning).