Summary: We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration version of an optimization problem \(Q\) takes as input two feasible solutions \(S\) and \(T\) and determines if there is a sequence of reconfiguration steps that can be applied to transform \(S\) into \(T\) such that each step results in a feasible solution to \(Q\). For most of the results in this paper, \(S\) and \(T\) are subsets of vertices of a given graph and a reconfiguration step adds or deletes a vertex. Our study is motivated by recent results establishing that for most NP-hard problems, the classical complexity of reconfiguration is PSPACE-complete.
We address the question for several important graph properties under two natural parameterizations: \(k\), the size of the solutions, and \(\ell \), the length of the sequence of steps. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter algorithms for the reconfiguration versions of Vertex Cover and, more generally, Bounded Hitting Set and Feedback Vertex Set, all parameterized by \(k\). In contrast, we show that reconfiguring Unbounded Hitting Set is W[2]-hard when parameterized by \(k + \ell \). We also demonstrate the W[1]-hardness of the reconfiguration versions of a large class of maximization problems parameterized by \(k + \ell \), and of their corresponding deletion problems parameterized by \(\ell \); in doing so, we show that there exist problems in FPT when parameterized by \(k\), but whose reconfiguration versions are W[1]-hard when parameterized by \(k + \ell \).
For the entire collection see [Zbl 1276.68019].