  We consider the problem of query-efficient global max-cut on a weighted
undirected graph in the value oracle model examined by [RSW18]. Graph
algorithms in this cut query model and other query models have recently been
studied for various other problems such as min-cut, connectivity,
bipartiteness, and triangle detection. Max-cut in the cut query model can also
be viewed as a natural special case of submodular function maximization: on
query $S \subseteq V$, the oracle returns the total weight of the cut between
$S$ and $V \backslash S$.
  Our first main technical result is a lower bound stating that a deterministic
algorithm achieving a $c$-approximation for any $c > 1/2$ requires $\Omega(n)$
queries. This uses an extension of the cut dimension to rule out approximation
(prior work of [GPRW20] introducing the cut dimension only rules out exact
solutions). Secondly, we provide a randomized algorithm with $\tilde{O}(n)$
queries that finds a $c$-approximation for any $c < 1$. We achieve this using a
query-efficient sparsifier for undirected weighted graphs (prior work of
[RSW18] holds only for unweighted graphs).
  To complement these results, for most constants $c \in (0,1]$, we nail down
the query complexity of achieving a $c$-approximation, for both deterministic
and randomized algorithms (up to logarithmic factors). Analogously to general
submodular function maximization in the same model, we observe a phase
transition at $c = 1/2$: we design a deterministic algorithm for global
$c$-approximate max-cut in $O(\log n)$ queries for any $c < 1/2$, and show that
any randomized algorithm requires $\Omega(n/\log n)$ queries to find a
$c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any
deterministic algorithm requires $\Omega(n^2)$ queries to find an exact max-cut
(enough to learn the entire graph).
