Processing math: 100%
Center for Algorithms and Theory of Computation

CS 269S, Winter 2018: Theory Seminar
Bren Hall, Room 1423, 1pm


February 9, 2018:

Linear-Time Algorithm for Sliding Tokens on Trees

Sid Gupta

Suppose that we are given two independent sets Ib and Ir of a graph such that |Ib|=|Ir|, and imagine that a token is placed on each vertex in Ib. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms Ib into Ir 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 Ib and Ir 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.)