  Consider a graph problem that is locally checkable but not locally solvable:
given a solution we can check that it is feasible by verifying all
constant-radius neighborhoods, but to find a solution each node needs to
explore the input graph at least up to distance $\Omega(\log n)$ in order to
produce its output. We consider the complexity of such problems from the
perspective of volume: how large a subgraph does a node need to see in order to
produce its output. We study locally checkable graph problems on bounded-degree
graphs. We give a number of constructions that exhibit tradeoffs between
deterministic distance, randomized distance, deterministic volume, and
randomized volume:
  - If the deterministic distance is linear, it is also known that randomized
distance is near-linear. In contrast, we show that there are problems with
linear deterministic volume but only logarithmic randomized volume.
  - We prove a volume hierarchy theorem for randomized complexity: among
problems with linear deterministic volume complexity, there are infinitely many
distinct randomized volume complexity classes between $\Omega(\log n)$ and
$O(n)$. This hierarchy persists even when restricting to problems whose
randomized and deterministic distance complexities are $\Theta(\log n)$.
  - Similar hierarchies exist for polynomial distance complexities: for any $k,
\ell \in N$ with $k \leq \ell$, there are problems whose randomized and
deterministic distance complexities are $\Theta(n^{1/\ell})$, randomized volume
complexities are $\Theta(n^{1/k})$, and whose deterministic volume complexities
are $\Theta(n)$.
  Additionally, we consider connections between our volume model and massively
parallel computation (MPC). We give a general simulation argument that any
volume-efficient algorithm can be transformed into a space-efficient MPC
algorithm.
