In Plato's Republic, he proposed his vision for a utopian city-state. Essentially, citizens would be ranked into castes: bronze/iron for laborers and artisans, silver for soldiers, gold for leaders. When people came of age in this utopia, they would be paired off through what they would be told is a random lottery.
In reality, however, the city's leaders would decide on the couplings ahead of time, pairing like with like. Plato called this the "noble lie", and it neatly demonstrates what any government-run dating system would probably develop into: a systemic eugenics program.