  We define a relative version of the Turaev-Viro invariants for an ideally
triangulated compact 3-manifold with non-empty boundary and a coloring on the
edges, generalizing the Turaev-Viro invariants [35] of the manifold. We also
propose the Volume Conjecture for these invariants whose asymptotic behavior is
related to the volume of the manifold in the hyperbolic polyhedral metric [22,
23] with singular locus the edges and cone angles determined by the coloring,
and prove the conjecture in the case that the cone angles are sufficiently
small. This suggests an approach of solving the Volume Conjecture for the
Turaev-Viro invariants proposed by Chen-Yang [8] for hyperbolic 3-manifolds
with totally geodesic boundary.
