Cyclic Shift Problems on Graphs

TL;DR

This paper introduces a new graph reconfiguration problem inspired by mechanical puzzles, analyzing the reachability and complexity of token rearrangements via cyclic shifts along specified cycles.

Contribution

It characterizes which configurations are reachable and provides efficient methods for token shifting, while proving the NP-hardness of finding shortest solutions.

Findings

Characterization of reachable configurations

Efficient algorithms for token shifting

NP-hardness of shortest sequence computation

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.