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 2log1− n/k hard to approximate for all constant > 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 ⊆ BPTIME(2polylog(n) ), we show that for every k ≥ 3 and every constant > 0 it is hard to approximate the basic k-spanner problem within a factor better than 2(log1− 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...

• Type: Journal
• Year: 2016
• Where: TALG
• Authors: Michael Dinitz, Guy Kortsarz, Ran Raz
• Keywords: TALG 2016