  Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of
Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci
flow on compact $3$-manifolds with boundary for finding complete hyperbolic
metrics on cusped $3$-manifolds. Dual to Casson and Rivin's program of
maximizing the volume of angle structures, combinatorial Ricci flow finds the
complete hyperbolic metric on a cusped $3$-manifold by minimizing the co-volume
of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may
develop singularities. We overcome this difficulty by extending the flow
through the potential singularities using Luo-Yang's extension. It is shown
that the existence of a complete hyperbolic metric on a cusped $3$-manifold is
equivalent to the convergence of the extended combinatorial Ricci flow, which
gives a new characterization of existence of a complete hyperbolic metric on a
cusped $3$-manifold dual to Casson and Rivin's program. The extended
combinatorial Ricci flow also provides an effective algorithm for finding
complete hyperbolic metrics on cusped $3$-manifolds.
