  The complexity of (unbounded-arity) Max-CSPs under structural restrictions is
poorly understood. The two most general hypergraph properties known to ensure
tractability of Max-CSPs, $\beta$-acyclicity and bounded (incidence) MIM-width,
are incomparable and lead to very different algorithms.
  We introduce the framework of point decompositions for hypergraphs and use it
to derive a new sufficient condition for the tractability of (structurally
restricted) Max-CSPs, which generalises both bounded MIM-width and
\b{eta}-acyclicity. On the way, we give a new characterisation of bounded
MIM-width and discuss other hypergraph properties which are relevant to the
complexity of Max-CSPs, such as $\beta$-hypertreewidth.
