We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of eciding whether $s$~pebbles suffice to reversibly pebble a DAG $G$ is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time.

翻译：本文研究了[Paterson and Hewitt '70]中提出的有界扇入有向无环图（DAG）上的卵石博弈，以及[Bennett '89]中提出的该博弈的可逆版本，并探讨了在这些博弈中精确或近似判定给定DAG所需卵石数量的难度。我们证明了判定$s$个卵石是否足以可逆地卵石覆盖DAG $G$的问题是PSPACE完全的，这与[Gilbert, Lengauer and Tarjan '80]中针对标准卵石博弈的结论一致。通过两种不同的图积构造，我们进一步强化了这些结果，证明标准卵石空间和可逆卵石空间在任意加法常数近似下均为PSPACE难问题。据我们所知，这是无限制设定下（即使对于多项式时间）卵石博弈近似难度的首批结果。此外，由于[Chan '13]证明了可逆卵石博弈等价于[Dymond and Tompa '85]和[Raz and McKenzie '99]中的博弈，我们的结果同样适用于Dymond-Tompa博弈和Raz-McKenzie博弈；由同一篇论文可知，解析深度在任意加法常数近似下均为PSPACE难判定问题。我们还得到了可逆卵石空间与标准卵石空间之间的乘法对数分离，这改进了此前已知的加法对数分离，且可能具有紧界性（尽管我们未能证明这一点）。我们留下了以下开放性问题：若将计算资源从多项式空间降低至更常见的多项式时间设定，我们的加法近似难度结果能否增强为乘法界。