Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs

TL;DR

This paper proves the difficulty of approximating the minimum pebbles needed in pebble games on DAGs, showing PSPACE-hardness and constructing graphs with exponential move complexity for constant pebbles.

Contribution

It provides a simplified proof of known hardness results and introduces explicit graphs with high move complexity, answering open questions in pebble game complexity.

Findings

Determined PSPACE-hardness for pebble game approximation within additive $n^{1/3-\\epsilon}$.

Constructed explicit DAG families requiring exponential moves with constant pebbles.

Answered open question on existence of DAGs with high move complexity for fixed pebbles.

Abstract

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].