  The observable macroscopic properties of relativistic stars (whose equations
of state are known) can be predicted by solving the stellar structure equations
that follow from Einstein's equation. For neutron stars, however, our knowledge
of the equation of state is poor, so the direct stellar structure problem can
not be solved without modeling the highest density part of the equation of
state in some way. This talk will describe recent work on developing a model
independent approach to determining the high-density neutron-star equation of
state by solving an inverse stellar structure problem. This method uses the
fact that Einstein's equation provides a deterministic relationship between the
equation of state and the macroscopic observables of the stars which are
composed of that material. This talk illustrates how this method will be able
to determine the high-density part of the neutron-star equation of state with
few percent accuracy when high quality measurements of the masses and radii of
just two or three neutron stars become available. This talk will also show that
this method can be used with measurements of other macroscopic observables,
like the masses and tidal deformabilities, which can (in principle) be measured
by gravitational wave observations of binary neutron-star mergers.
