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 [4] and strengthen the result to show that it is PSPACE-hard to determine the minimum number of pebbles to an additive n~(1/3-ε) term for all ε > 0, which improves upon the currently known additive constant hardness of approximation [4] 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 0 < k < √n pebbles requires Ω((n/k)~k) moves to pebble in both the standard and black-white pebble games. This independently answers an open question summarized in [14] 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 [1].
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