Biodegrading releases the carbon as CO2 to the atmosphere. With our current concerns over global warming, not biodegrading isn't a bad short term solution till we get a handle on global warming.
Depending on what the waste is, can it be recycled? If not, can it be used as an alternative fuel source? Can it be made into something else?
Many wastes do not naturally degrade but can be used for something else or reused in its current form.
If not recyclable then incinerate.
easy
recycle them
hope i helped :)