  We design a Local Computation Algorithm (LCA) for the set cover problem.
Given a set system where each set has size at most $s$ and each element is
contained in at most $t$ sets, the algorithm reports whether a given set is in
some fixed set cover whose expected size is $O(\log{s})$ times the minimum
fractional set cover value. Our algorithm requires $s^{O(\log{s})} t^{O(\log{s}
\cdot (\log \log{s} + \log \log{t}))}$ queries. This result improves upon the
application of the reduction of [Parnas and Ron, TCS'07] on the result of [Kuhn
et al., SODA'06], which leads to a query complexity of $(st)^{O(\log{s} \cdot
\log{t})}$.
  To obtain this result, we design a parallel set cover algorithm that admits
an efficient simulation in the LCA model by using a sparsification technique
introduced in [Ghaffari and Uitto, SODA'19] for the maximal independent set
problem. The parallel algorithm adds a random subset of the sets to the
solution in a style similar to the PRAM algorithm of [Berger et al., FOCS'89].
However, our algorithm differs in the way that it never revokes its decisions,
which results in a fewer number of adaptive rounds. This requires a novel
approximation analysis which might be of independent interest.
