Parameterized inapproximability: From Clique to PIH

Parameterized approximation, first proposed by Mike Fellows, approaches NP-hard problems by allowing the running time of an approximation algorithm to be superpolynomial in the parameter of an problem instance yet still polynomial in the size of the instance itself. One of the main open questions in the area is whether we can approximate the parameterized clique problem within some nontrivial ratio. It is also conjectured by Fellows that no such algorithms exist. In this article, we explain some recent progress on this question.

Similarly to the classical polynomial time inapproximability of the clique problem, the constraint satisfaction problem, i.e., CSP, plays a key role in most of the known inapproximability results of the parameterized clique problem. As a matter of fact, the parameterized inapproximability hypothesis, i.e., PIH, concerning the binary CSP has been long believed as a viable path towards the inapproximability of the parameterized clique problem. Although it turns out that those recent results do not rely on PIH, the method discovered for the parameterized clique problem leads to a proof of a version of PIH under the exponential time hypothesis, which we will also explain in this article.