The Complexity of Bounded Length Graph Recoloring

We study the following question: Given are two $k$-colorings $\alpha$ and $\beta$ of a graph $G$ on $n$ vertices, and integer $\ell$. The question is whether $\alpha$ can be modified into $\beta$, by recoloring vertices one at a time, while maintaining a $k$-coloring throughout, and using at most $\ell$ such recoloring steps. This problem is weakly PSPACE-hard for every constant $k\ge 4$. We show that it is also strongly NP-hard for every constant $k\ge 4$. On the positive side, we give an $O(f(k,\ell) n^{O(1)})$ algorithm for the problem, for some computable function $f$. Hence the problem is fixed-parameter tractable when parameterized by $k+\ell$. Finally, we show that the problem is W[1]-hard (but in XP) when parameterized only by $\ell$.