In the previous chapters we surveyed many well-known algorithmic techniques and successfully applied them to obtain efficient algorithmic solutions for problems from varied domains. Yet, we cannot claim that there is some general methodology for obtaining efficient algorithms for any given problem. To the contrary any new problem often presents unknown challenges that require new insights and the question that is uppermost in anyone's mind is - what are the problems that are notoriously difficult? We need to first set some target before we can assign a notion of difficulty to a problem. For reasons that will become clear later on, the quest is to design a polynomial time algorithm for any given well-defined problem. This may appear too liberal initially since by definition even n100 is a polynomial. However, even this has proved elusive for a large number of problems among which we have come across one in the preceding chapters, namely, 0-1 Knapsack problem. Despite promising starts, we could never claim a truly polynomial time algorithm.

In addition to fixing the limits of practical computation being a polynomial time algorithm, we need to specify the underlying computational model since that will also affect what is achievable in polynomial time. Fortunately, the notion of polynomial time is a robust concept that is not significantly affected by the choice of computational model, except for some constant factors in the exponent of n. We will discuss about a large class of natural and important problems that admits a characterization, that is very intuitive and has resulted in a very interesting theoretical framework. Since the reader is familiar with the Knapsack problem, we will illustrate this framework in the context of the Knapsack problem. Consider a Prover, Verifier interactive game regarding any given Knapsack instance. The Prover is trying to convince the Verifier that she has an efficient (polynomial-time) algorithm without actually revealing the technique. For proof, given an instance of Knapsack, she provides a list of objects chosen in the optimal solution to the Verifier who can easily verify the feasibility and the total profit that can be obtained from this solution.