Label cover instances with large girth and the hardness of approximating basic k-spanner

We study the well-known Label Cover problem under the additional requirement that problem instances have large girth. We show that if the girth is some k, the problem is roughly 2^(log1-εn)/k hard to approximate for all constant ε > 0. A similar theorem was claimed by Elkin and Peleg [ICALP 2000] as part of an attempt to prove hardness for the basic k-spanner problem, but their proof was later found to have a fundamental error. Thus we give both the first non-trivial lower bound for the problem of Label Cover with large girth as well as the first full proof of strong hardness for the basic k-spanner problem, which is both the simplest problem in graph spanners and one of the few for which super-logarithmic hardness was not known. Assuming NP ⊈ BPTIME(2^polylog(n)), we show (roughly) that for every k ≥ 3 and every constant ε > 0 it is hard to approximate the basic k-spanner problem within a factor better than 2 ^(log1-εn)/k. This improves over the previous best lower bound of only Ω(logn)/k from [17]. Our main technique is subsampling the edges of 2-query PCPs, which allows us to reduce the degree of a PCP to be essentially equal to the soundness desired. This turns out to be enough to basically guarantee large girth.