Cyclic Shift Problems on Graphs
Abstract
We study a new reconfiguration problem inspired by classic mechanical puzzles: a colored token is placed on each vertex of a given graph; we are also given a set of distinguished cycles on the graph. We are tasked with rearranging the tokens from a given initial configuration to a final one by using cyclic shift operations along the distinguished cycles. We first investigate a large class of graphs, which generalizes several classic puzzles, and we give a characterization of which final configurations can be reached from a given initial configuration. Our proofs are constructive, and yield efficient methods for shifting tokens to reach the desired configurations. On the other hand, when the goal is to find a shortest sequence of shifting operations, we show that the problem is NP-hard, even for puzzles with tokens of only two different colors.
Keywords
Cite
@article{arxiv.2009.10981,
title = {Cyclic Shift Problems on Graphs},
author = {Kwon Kham Sai and Ryuhei Uehara and Giovanni Viglietta},
journal= {arXiv preprint arXiv:2009.10981},
year = {2020}
}