  We prove complex contraction for zero-free regions of counting weighted set
cover problem in which an element can appear in an unbounded number of sets,
thus obtaining fully polynomial-time approximation schemes(FPTAS) via
Barvinok's algorithmic paradigm\cite{barvinok2016combinatorics}. Relying on the
computation tree expansion, our approach does not need proof of correlation
decay in the real axis. We directly look in the complex plane for a region that
contracts into its interior as the tree recursion procedure goes from leaves to
the root.
  For the class of problems under the framework of weighted set covers, we are
able to give a general approach for describing the contraction regions and draw
a unified algorithmic conclusion. Several previous results, including counting
(weighted-)edge covers, counting bipartite independent sets and counting
monotone CNFs can be completely or partially covered by our main theorem. In
contrast to the correlation decay method which also depends on tree expansions
and needs different potential functions for different problems, our approach is
more generic in the sense that our contraction region for different problems
shares a common shape in the complex plane.
