  This work extends monotonicity-based methods in inverse problems to the case
of the Helmholtz (or stationary Schr\"odinger) equation $(\Delta + k^2 q) u =
0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued
scattering coefficient function $q$. We show a monotonicity relation between
the scattering coefficient $q$ and the local Neumann-Dirichlet operator that
holds up to finitely many eigenvalues. Combining this with the method of
localized potentials, or Runge approximation, adapted to the case where
finitely many constraints are present, we derive a constructive
monotonicity-based characterization of scatterers from partial boundary data.
We also obtain the local uniqueness result that two coefficient functions $q_1$
and $q_2$ can be distinguished by partial boundary data if there is a
neighborhood of the boundary where $q_1\geq q_2$ and $q_1\not\equiv q_2$.
