Linear-Time Algorithm for Sliding Tokens on Trees

TL;DR

This paper presents a linear-time algorithm for solving the sliding tokens problem on trees, providing efficient decision and sequence construction methods, and analyzing sequence length requirements.

Contribution

It introduces the first linear-time decision algorithm for sliding tokens on trees and constructs quadratic-length sequences, advancing understanding of token sliding complexity.

Findings

Decision problem solvable in linear time on trees

Constructs quadratic-length sequences between token configurations

Identifies instances requiring quadratic sequence length on paths

Abstract

Suppose that we are given two independent sets $I_b$ and $I_r$ of a graph such that $|I_b|=|I_r|$, and imagine that a token is placed on each vertex in $I_b$. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms $I_b$ into $I_r$ so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between $I_b$ and $I_r$ whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.