  We provide a systematic deterministic numerical scheme to approximate the
volume (i.e. the Lebesgue measure) of a basic semi-algebraic set whose
description follows a sparsity pattern. As in previous works (without
sparsity), the underlying strategy is to consider an infinite-dimensional
linear program on measures whose optimal value is the volume of the set. This
is a particular instance of a generalized moment problem which in turn can be
approximated as closely as desired by solving a hierarchy of semidefinite
relaxations of increasing size. The novelty with respect to previous work is
that by exploiting the sparsity pattern we can provide a sparse formulation for
which the associated semidefinite relaxations are of much smaller size. In
addition, we can decompose the sparse relaxations into completely decoupled
subproblems of smaller size, and in some cases computations can be done in
parallel. To the best of our knowledge, it is the first contribution that
exploits sparsity for volume computation of semi-algebraic sets which are
possibly high-dimensional and/or non-convex and/or non-connected.
