The Complexity of Bounded Length Graph Recoloring and CSP Reconfiguration

Abstract

In the first part of this work we study the following question: Given two $k$-colorings $\alpha$ and $\beta$ of a graph $G$ on $n$ vertices and an integer $\ell$, can $\alpha$ 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 the problem is also strongly NP-hard for every constant $k \ge 4$ and W[1]-hard (but in XP) when parameterized only by $\ell$. On the positive side, we show that the problem is fixed-parameter tractable when parameterized by $k+\ell$. In fact, we show that the more general problem of $\ell$-length bounded reconfiguration of constraint satisfaction problems (CSPs) is fixed-parameter tractable parameterized by $k+\ell +r$, where $r$ is the maximum constraint arity and $k$ is the maximum domain size. We show that for parameter $\ell$, the latter problem is W[2]-hard, even for $k=2$. Finally, if $p$ denotes the number of variables with different values in the two given assignments, we show that the problem is W[2]-hard when parameterized by $\ell -p$, even for $k=2$ and $r=3$.

Amer E. Mouawad, Naomi Nishimura— Research supported by the Natural Science and Engineering Research Council of Canada.