  Pebble games are single-player games on DAGs involving placing and moving
pebbles on nodes of the graph according to a certain set of rules. The goal is
to pebble a set of target nodes using a minimum number of pebbles. In this
paper, we present a possibly simpler proof of the result in [CLNV15] and
strengthen the result to show that it is PSPACE-hard to determine the minimum
number of pebbles to an additive $n^{1/3-\epsilon}$ term for all $\epsilon >
0$, which improves upon the currently known additive constant hardness of
approximation [CLNV15] in the standard pebble game. We also introduce a family
of explicit, constant indegree graphs with $n$ nodes where there exists a graph
in the family such that using constant $k$ pebbles requires $\Omega(n^k)$ moves
to pebble in both the standard and black-white pebble games. This independently
answers an open question summarized in [Nor15] of whether a family of DAGs
exists that meets the upper bound of $O(n^k)$ moves using constant $k$ pebbles
with a different construction than that presented in [AdRNV17].
