Well, icing reduces swelling. Swelling caused by fluid incursion causes tissue damage and scar tissue, plus it takes time to heal. Ice for a maximum of 5 minutes but do not allow the tissues to become chilled through - the idea is to cool, not to freeze (even worse). Repeat only a couple of times per hour, and for only a few hours in the days after the injury. Don't heat, or alternate. Heat will promote swelling through fluid flow, the exact opposite of what you want.

I can thoroughly recommend a TENS apparatus for rapid healing of sprains, I have the OMRON E3 unit which you should be able to hire or buy at relatively low cost. It has helped me get past a muscle tear about 14 days before a race, and while I didn't perform at my peak, I could in fact run and didn't cause any further damage. Highly recommended as the wonder solution.