A note on degree vs gap of Min-Rep Label Cover and improved inapproximability for connectivity problems

Abstract

This note concerns the trade-off between the degree of the constraint graph and the gap in hardness of approximating the Min-Rep variant of Label Cover (aka Projection Game). We make a very simple observation that, for NP-hardness with gap g, the degree can be made as small as O(g log g), which improves upon the previousÕ(g 1/2 ) bound from a work of Laekhanukit [Lae14]. Note that our bound is optimal up to a logarithmic factor since there is a trivial ∆-approximation for Min-Rep where ∆ is the maximum degree of the constraint graph.Thanks to known reductions [CLNV14, CK12, CMVZ16, Lae14], this improvement implies better hardness of approximation results for Rooted k-Connectivity, Vertex-Connectivity Survivable Network Design and Vertex-Connectivity k-Route Cut. *

Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity

“…In addition, we obtain an approximation hardness exponential in k by setting a different parameter in the reduction, which improves upon the previously known approximation hardness of Ω (k/ log k) due to Manurangsi [Man19] (which is in turn based on the two previous results [Lae14, CLNV14]), and is the first known approximation hardness for connectivity problems whose ratio is exponential in the connectivity requirement.…”

“…Manurangsi [Man19] proved that the label cover problem has a hardness gap in terms of the maximum degree of G.…”

Survivable Network Design Revisited: Group-Connectivity

“…In fact, the hardness result holds even when all the connectivity requirements are t0, 1u. The bounds have been improved in the subsequent works to n op1q under Gap-ETH in the work of Dinur [Din16] and has also been refined in [CLNV14,Lae14,DM18,Man19,LCLZ22]. To date, we know that even in the special case of single-source k-connectivity, the problem is at least as hard as the Label-Cover problem [CLNV14], and in the very recent work, Liao, Chen, Laekhanukit and Zhang showed that Directed-SNDP admits no non-trivial approximation algorithms unless NP " ZPP.…”

“…That is, no opq{ log qq-approximation algorithm exists unless NP " ZPP, and no opqq-approximation algorithm exists unless the Strongish Planted Clique Hypothesis is false [LCLZ22]. When focusing on the hardness ratio in terms of k, it is shown in [CGL15] (combined with [Lae14] and the improvement in [Man19]) that the approximation hardness is k 1{5´ , for ą 0, assuming NP " ZPP.…”

“…The factor k appears quite naturally in network design problems and might be the approximability threshold. However, the best known negative result still has a lower bound of k 1{5´ , for ą 0, assuming NP ‰ ZPP, [CDE `18] (combined with [Lae14] and the improvement in [Man19]). While it is rather unnatural that a survivable network design problem would have a hardness factor beyond k, the recent result of Liao, Chen, Laekhanukit and Zhang [LCLZ22] shows that this is the case for the sister problem of Group EC-SNDP, namely the k-connected directed Steiner tree problem (k-DST) (a.k.a, directed single-source k-connectivity).…”