Reduction of swelling does not promote healing. It promotes comfort and ease of use which (specifically in the sports medicine community) is the more pressing concern for injuries that are not considered debilitating or serious enough to refrain from the specific activity that caused them.
Someone on the thread earlier mentioned swelling as blood rushing to the affected area; this is inaccurate.
The process of acute inflammation is initiated by cells already present in all tissues, mainly resident macrophages, dendritic cells, histiocytes, Kuppfer cells and mastocytes. At the onset of an infection, burn, or other injuries, these cells undergo activation and release inflammatory mediators responsible for the clinical signs of inflammation.
Vasodilation and its resulting increased blood flow causes the redness] and increased heat. Increased permeability of the blood vessels results in an exudation (leakage) of plasma proteins and fluid into the tissue, which manifests itself as swelling.
Some of the released mediators such as bradykinin increase the sensitivity to pain. The mediator molecules also alter the blood vessels to permit the migration of leukocytes, mainly neutrophils, outside of the blood vessels into the tissue. The neutrophils migrate along a chemotactic gradient created by the local cells to reach the site of injury.
The repair process involves the regeneration of tissue or neo-genesis when tissue is destroyed. Once the repair process is finished swelling tends to dissipate on it’s own due to a lack of production of the chemotactic response.
So while reducing the swelling may make it easier to resume your activities, it won’t speed up or promote healing. Giving your body the opportunity to heal can be a good thing.