Nearly Optimal NP-Hardness of Unique Coverage
Venkatesan Guruswami, Euiwoong Lee · SIAM Journal on Computing · 2017
The Unique Coverage problem, given a universe $V$ of elements and a collection $E$ of subsets of $V$, asks to find $S \subseteq V$ to maximize the number of $e \in E$ that intersects $S$ in exactly one element. When each $e \in E$ has cardinality at most $k$, it is also known as 1-in-$k$ Hitting Set and admits a simple $\Omega(\frac{1}{\log k})$-approximation algorithm. For constant $k$, we prove that 1-in-$k$ Hitting Set is NP-hard to approximate within a factor $O(\frac{1}{\log k})$. This improves the result of Guruswami and Zhou [Theory Comput., 8 (2012), pp. 239--267], who proved the same result assuming the Unique Games Conjecture. For Unique Coverage, we prove that it is hard to approximate within a factor $O(\frac{1}{\log^{1 - \epsilon} n})$ for any $\epsilon > 0$, unless NP admits quasi-polynomial time algorithms. This improves the results of Demaine et al. [SIAM J. Comput., 38 (2008), pp. 1464--1483], including their $\approx 1/\log^{1/3} n$ inapproximability factor, which was proven under the Random 3SAT Hypothesis. Our simple proof combines ideas from two classical inapproximability results for the Set Cover and Constraint Satisfaction Problems, made efficient by various derandomization methods based on bounded independence.