On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes

In the (k, h)-SetCover problem, we are given a collection S of sets over a universe U, and the goal is to distinguish between the case that S contains k sets which cover U, from the case that at least h sets in S are needed to cover U. Lin (ICALP’19) recently showed a gap creating reduction from the (k, k + 1)-SetCover problem on universe of size O_k(log |S|) to the (k, ^qk_log^log_log^|S|_|S| · k ) -SetCover problem on universe of size |S|. In this paper, we prove a more scalable version of his result: given any error correcting code C over alphabet [q], rate ρ, and relative distance δ, we use C to create a reduction from the (k, k + 1)-SetCover problem on universe U to the (k, ^2qk₁₋²_δ) -SetCover problem on universe of size ^log_ρ^|S| |U|^qk. Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code C. We further show that one can recover the precise result of Lin by using a code which also achieves optimal parameters as a perfect hash function.