There is a class of problems known as NP-complete, as well as a similar class called NP-hard that are at least as difficult; for this discussion you can consider those terms interchangeable. These problems have the property that if any of them can be solved efficiently (that is, in polynomial time), then they all can be. Further, despite much effort by many smart people, no efficient algorithm has ever been devised for any of them. This leads to the prevailing opinion that none of the NP-complete/NP-hard problems can be efficiently solved. You can know that a problem you face is NP-hard if a magic oracle that solved your problem instantly could be used to solve another problem that is known to be NP-hard in polynomial time (hundreds of such problems are known). If this is true, you have a number of options (which I will elaborate upon in future posts), but the most likely one is to devise a heuristic that will produce [hopefully] good, but not necessarily optimal, solutions.
(Aside: The title of this post might be my most obscure ever; +1 to anyone who can identify it without consulting the web.)
Labels: algorithms, np-completeness, theory