Complete enumeration and shortcut and balanced overlap and such are algorithms Sawtooth Software uses for making designs. There are also other strategies for making efficient designs through computer searches and methods for making designs using orthogonal design catalogs. Those are all just WAYS of making designs.
An alternative-specific design is a particular type of design one does for a very specific reason. In fact, there are three types of alternative-specific designs.
(1) The type with the greatest amount of alternative specificity is a design where different alternatives can have different attributes. The classic example here is a transportation study where someone might walk to work, take a bus or drive to work. If you take a bus you have a "wait time" attribute that you won't have if you walk or drive. If you drive you have a "petrol cost" attribute that you won't have if you walk or take the bus, and a parking cost, too, for that matter. There would also be attributes you might share, like travel time.
(2) Another type of alternative-specific design happens when your different alternatives can have different levels for the same attribute. For the travel example above, walking might have travel times of 15 minutes, 30 minutes and 45 minutes while taking the bus might have times of 5, 10 and 15 minutes. If your study is of cold medications, your "package size" attribute might be measured in number of tablets for swallowable pill medications but in ml for liquid medications.
(3) Finally, you might have alternative-specific sensitivities even when alternatives share the same levels for their attributes. For example, you might have prices of $1.00, $1.50 and $1.50 for Coke and for Pepsi, but if you think the two brands might have different price sensitivity you could estimate them as alternative-specific effects. Of course you might also use interactions in this case to get the same result.
As you can see, alternative specific effects often use what are called "labeled designs," where one of the attributes (usually brand) determines which other attributes might be alternative-specific.
I hope this helps.