  The low-energy nuclear structure and two-neutrino double-$\beta$
($2\nu\beta\beta$) decay are studied within the interacting boson model (IBM)
that is based on the nuclear energy density functional (EDF). The IBM
Hamiltonian describing the initial and final even-even nuclei, and the
interacting boson fermion-fermion Hamiltonian producing the intermediate states
of the neighboring odd-odd nuclei, are determined by the microscopic inputs
provided by the self-consistent mean-field (SCMF) calculations employing a
relativistic EDF and a separable pairing force. Sensitivities of the low-lying
structure and $2\nu\beta\beta$-decay properties to the pairing strength are
specifically analyzed. It is shown that the SCMF calculations with the
decreased and increased pairing strengths lead to the quadrupole-quadrupole
interaction strengths in the IBM that are, respectively, significantly enhanced
and reduced in magnitude. When the increased pairing is adopted, in particular,
the energy levels of the excited $0^+$ states are lowered, and the predicted
$2\nu\beta\beta$-decay nuclear matrix elements (NMEs) increase in magnitude
systematically. The mapped IBM employing the increased pairing force generates
effective NMEs and half-lives that are overall in a reasonable agreement with
the experimental data.
