# Linear-Time Algorithm for Sliding Tokens on Trees

## Sid Gupta

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 talk, 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.

(Based on a paper from ISAAC 2014 and Theoretical Computer Science 2015 by Demaine, Demaine, Fox-Epstein, Hoang, Ito, Ono, Otachi, Uehara, and Yamada.)