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

• Authors: Michael Dinitz;Guy Kortsarz;Ran Raz
• Venue: ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
• Year: 2012

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 ε0. 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 $NP \not\subseteq BPTIME(2^{polylog(n)})$, we show (roughly) that for every k≥3 and every constant ε0 it is hard to approximate the basic k-spanner problem within a factor better than $2^{(\log^{1-\epsilon} n) / k}$. 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.