Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada, “Linear-time algorithm for sliding tokens on trees”, in Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC 2014), Lecture Notes in Computer Science, volume 8889, December 15–17, 2014, pages 389–400.

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 show that the problem is solvable for trees in quadratic time. Our proof is constructive: for a yes-instance, we can find an actual sequence of independent sets between I_b and I_r whose length (i.e., the number of token-slides) is quadratic. We note that there exists an infinite family of instances on paths for which any sequence requires quadratic length.

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