A Note on Degree vs Gap of Min-Rep Label Cover and Improved Inapproximability for Connectivity Problems
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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 g), which improves upon the previous Õ(g^1/2) bound from a work of Laekhanukit (SODA'14). Note that our bound is optimal up to a logarithmic factor since there is a trivial Δ-approximation for Min-Rep where Δ is the maximum degree of the constraint graph. Thanks to known reductions, this improvement implies better hardness of approximation results for Rooted k-Connectivity, Vertex-Connectivity Survivable Network Design and Vertex-Connectivity k-Route Cut.
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