Polynomial Turing Kernels for Clique with an Optimal Number of Queries

A polynomial Turing kernel for some parameterized problem $P$ is a polynomial-time algorithm that solves $P$ using queries to an oracle of $P$ whose sizes are upper-bounded by some polynomial in the parameter. Here the term "polynomial" refers to the bound on the query sizes, as the running time of any kernel is required to be polynomial. One of the most important open goals in parameterized complexity is to understand the applicability and limitations of polynomial Turing Kernels. As any fixed-parameter tractable problem admits a Turing kernel of some size, the focus has mostly being on determining which problems admit such kernels whose query sizes can be indeed bounded by some polynomial. In this paper we take a different approach, and instead focus on the number of queries that a Turing kernel uses, assuming it is restricted to using only polynomial sized queries. Our study focuses on one the main problems studied in parameterized complexity, the Clique problem: Given a graph $G$ and an integer $k$, determine whether there are $k$ pairwise adjacent vertices in $G$. We show that Clique parameterized by several structural parameters exhibits the following phenomena: - It admits polynomial Turing kernels which use a sublinear number of queries, namely $O(n/\log^c n)$ queries where $n$ is the total size of the graph and $c$ is any constant. This holds even for a very restrictive type of Turing kernels which we call OR-kernels. - It does not admit polynomial Turing kernels which use $O(n^{1-\epsilon})$ queries, unless NP$\subseteq$coNP/poly. For proving the second item above, we develop a new framework for bounding the number of queries needed by polynomial Turing kernels. This framework is inspired by the standard lower bounds framework for Karp kernels, and while it is quite similar, it still requires some novel ideas to allow its extension to the Turing setting.