Nearly Optimal NP-Hardness of Unique Coverage

The Unique Coverage problem, given a universe V of elements and a collection E of subsets of V , asks to find S ⊆V to maximize the number of e ∈ E that intersects S in exactly one element. When each e ∈ E has cardinality at most k, it is also known as 1-in-k Hitting Set, and admits a simple Ω( 1 logk )-approximation algorithm. For constant k, we prove that 1-in-k Hitting Set is NP-hard to approximate within a factor O( 1 logk ). This improves the result of Guruswami and Zhou [SODA’11, ToC’12], who proved the same result assuming the Unique Games Conjecture. For Unique Coverage, we prove that it is hard to approximate within a factor O( 1 log1−ε n ) for any ε > 0, unless NP admits quasipolynomial time algorithms. This improves the results of Demaine et al. [SODA’06, SICOMP’08], including their ≈ 1/ log1/3 n inapproximability factor which was proven under the Random 3SAT Hypothesis. Our simple proof combines ideas from two classical inapproximability results for Set Cover and Constraint Satisfaction Problem, made efficient by various derandomization methods based on bounded independence. ∗Supported in part by NSF grant CCF-1115525. [email protected] †Supported by a Samsung Fellowship and NSF CCF-1115525. [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 155 (2015)