In this article, we will survey the new algorithm design method called holographic algorithms. This method uses perfect matching as a basic coding technique to encode computations, and then the FKT-algorithm to carry out the final computation. A particularly innovative idea is to choose a set of linear basis vectors to express and interpret a desired computation. In effect the algorithm is designed to manipulate sums of perfect matchings in superpositions, while the speed up is achieved by cancellations among such "holographic mixes". These holographic algorithms are quite unlike anything before, except perhaps quantum algorithms. At the heart of the computation is a process of introducing and then canceling exponentially many computational fragments. But unlike quantum algorithms, these holographic algorithms produce classical polynomial time algorithms. So far this method has produced some exotic algorithms for problems which were not known to be in P previously, and minor variations of which are known to be NP-complete or NP-hard.
The most intriguing question is whether this new theory can lead to any collapse of complexity classes. We contend that our belief of NP != P is based on the sense and experience that the usual algorithmic paradigms are insufficient for NP-hard problems (we don't have strong lower bounds for general models of computation). But does our erstwhile experience apply to these new exotic algorithms? If the answer is no, then it is conceivable that the new methodology may lead to a radically revised conception of P vs. NP. Of course it is quite possible that the theory of holographic algorithms does not in the end lead to any collapse of complexity classes. But even in this eventuality, as Valiant suggested in [54], "any proof of P != NP may need to explain, and not only to imply, the unsolvability" of NP-hard problems using this approach.