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 $2^{\log^{1-\epsilon} n/k}$ hard to approximate for all constant $\epsilon > 0$. A similar theorem was claimed by Elkin and Peleg [ICALP 2000], but their proof was later found to have a fundamental error. We use the new proof to show inapproximability 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 $NP \not\subseteq BPTIME(2^{polylog(n)})$, we show that for every $k \geq 3$ and every constant $\epsilon > 0$ it is hard to approximate the basic $k$-spanner problem within a factor better than $2^{(\log^{1-\epsilon} n) / k}$ (for large enough $n$). A similar hardness for basic $k$-spanner was claimed by Elkin and Peleg [ICALP 2000], but the error in their analysis of Label Cover made this proof fail as well. Thus for the problem of Label Cover with large girth we give the first non-trivial lower bound. For the basic $k$-spanner problem we improve the previous best lower bound of $\Omega(\log n)/k$ by Kortsarz [Algorithmica 1998]. 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 essentially guarantee large girth.