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Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

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 [4] and strengthen the result to show that it is PSPACE-hard to determine the minimum number of pebbles to an additive \(n^{1/3-\epsilon }\) term for all \(\epsilon > 0\), 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 \(0< k < \sqrt{n}\) pebbles requires \(\varOmega ((n/k)^k)\) 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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MIT CSAILCambridgeUSA

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