Hardness of Approximation in PSPACE and Separation Results for Pebble Games

Abstract

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 deciding whether s pebbles suffice to reversibly pebble a DAG G is PSPACEcomplete, 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. In the pebble game first studied by Paterson and Hewitt [26], one starts with an empty directed acyclic graph (DAG) G with bounded fan-in (and which in this paper in addition will always have a single sink) and places pebbles on the vertices according to the following rules:• If all (immediate) predecessors of an empty vertex v contain pebbles, a pebble may be placed on v.• A pebble may be removed from any vertex at any time. The goal is to get a pebble on the sink vertex of G with all other vertices being empty, and to do so while minimizing the total number of pebbles on G at any given time (the pebbling price of G). This game models computations with execution independent of the actual input. A pebble on a vertex indicates that the corresponding value is currently kept in memory and the objective is to perform the computation with the minimum amount of memory.The pebble game has been used to study flowcharts and recursive schemata [26] Bennett [3] introduced the reversible pebble game as part of a broader program [2] to investigate possibilities to eliminate (or significantly reduce) energy dissipation in logical computation. Another reason reversible computation is of interest is that observation-free quantum computation is inherently reversible. In the reversible pebble game, the moves performed in reverse order should also constitute a legal pebbling, whi...

Optimizing Quantum Space Using Spooky Pebble Games

“…In this section, we show that deciding whether a DAG can be spooky pebbled using P pebbles is PSPACE-complete answering the open question of [11]. For the irreversible [17] and reversible [18] pebble game, this problem is already known to be PSPACE-complete.…”

Red-Blue Pebble Game

“…the vertex already contains a blue pebble.) (4) No pebbles can be deleted from a vertex. As usual, red pebbles represent fast memory and blue pebbles represent slow memory; we assume that we have infinitely large slow memory, but only a bounded fast memory.…”

“…(3) The player can recolor a blue pebble to red, or a red pebble to blue. (4) The player can delete a pebble from a node at any time. Goal: Pebble all sink nodes with blue pebbles.…”

“…Despite somewhat extensive research on the upper and lower bounds of optimally pebbling a DAG in pebble games, the complexity of finding a minimum solution has fewer results. In fact, it is not yet known whether it is hard to find the minimum number of pebbles within a constant or logarithmic multiplicative approximation factor [4,8]. It turns out that finding a strategy to optimally pebble a graph in the standard pebble game is computationally difficult even when each vertex is allowed to be pebbled only once.…”

Reversible Pebbling Game for Quantum Memory Management

“…The problem complexity has been studied in [15] where the authors prove that the problem is PSPACE-complete, as the non-reversible pebbling game. An explicit asymptotic expression for the best time-space product is given in [16], while the asymptotic behavior on trees is studied in [17].…”