Noise cancellation never works on scales larger than the sound's wavelength, so it's only useful for headphones.
Modifying the hall physically is the only real solution to this problem.
Short of that, the first thing to try is indeed to bring the sound as directly to the audience as possible, by using a suitable PA arrangement. Big, single speakers located "out of sight" are generally worse than multiple speakers close to the audience, either at different levels (delay lines) or as a stack (approximate line array, which partially cancels the sound that goes to the ceiling).
If that's not enough, the only thing you can do is modify the sound signals itself. Obviously, a small acoustic ensemble will have less of a problem with reverb than a large, loud rock band. In particular, uncompressed drum accents cause reverb throughout the frequency spectrum that often overpowers everything else. So you want to make the drums themselves as quiet as possible. The monitor levels should also be kept down, because those signals arrive at the audience only indirectly.
Then, on the PA, you should aim at a strongly frequency-separated mix and rather agressive compression. This way, all instruments should come through reasonably well even when the time-domain is completely smeared out by the reverb.