On hardness of approximating the parameterized clique problem

In the Gap-clique(k; k/2 ) problem, the input is an n-vertex graph G, and the goal is to decide whether G contains a clique of size k or contains no clique of size k/2 . It is an open question in the study of fixed parameterized tractability whether the Gap-clique(k; k/2 ) problem is fixed parameter tractable, i.e., whether it has an algorithm that runs in time f(k) n, where f(k) is an arbitrary function of the parameter k and the exponent is a constant independent of k. In this paper, we give some evidence that the problem Gap-clique(k; k/2 ) is not fixed parameter tractable. Speciff-cally, we define a constraint satisfaction problem, which we call Deg-2-sat, where the input is a system of k0 quadratic equations in k0 variables over a finite field F of size n0, and the goal is to decide whether there is a solution in F that satisfies all the equations simultaneously. The main result in this paper is an "FPT-reduction" from Deg-2-sat to the Gap-clique(k; k/2 ) problem. If one were to hypothesize that the Deg-2-sat problem is not fixed parameter tractable, then our reduction would imply that the Gap-clique(k; k/2 ) problem is not fixed parameter tractable either. The reduction relies on the algebraic techniques used in proof of the PCP theorem.