On the Hardness of Red-Blue Pebble Games

TL;DR

This paper investigates the computational complexity of red-blue pebble games, demonstrating NP-hardness across variants and establishing limitations on approximation algorithms, highlighting the challenges in modeling and optimizing memory hierarchies.

Contribution

The paper introduces new hardness results for various red-blue pebble game models, including a novel variant, and analyzes approximation limits in the oneshot model.

Findings

Red-blue pebbling is NP-hard in all studied variants.

A δ-approximation for δ<2 in the oneshot model implies the falsehood of the unique games conjecture.

Greedy algorithms can perform significantly worse than optimal solutions.

Abstract

Red-blue pebble games model the computation cost of a two-level memory hierarchy. We present various hardness results in different red-blue pebbling variants, with a focus on the oneshot model. We first study the relationship between previously introduced red-blue pebble models (base, oneshot, nodel). We also analyze a new variant (compcost) to obtain a more realistic model of computation. We then prove that red-blue pebbling is NP-hard in all of these model variants. Furthermore, we show that in the oneshot model, a $\delta$-approximation algorithm for $\delta<2$ is only possible if the unique games conjecture is false. Finally, we show that greedy algorithms are not good candidates for approximation, since they can return significantly worse solutions than the optimum.