On the Parameterized Complexity of Reconfiguration Problems - Mouawad, Amer E.; Nishimura, Naomi; Raman, Venkatesh; Simjour, Narges; Suzuki, Akira (Algorithmica, 2017)
We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, 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 sets of vertices of a given graph and a reconfiguration step adds or removes a vertex. Our study is motivated by results establishing that for many 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 , a bound on the size of solutions, and ℓ , a bound on the length of reconfiguration sequences. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter tractable algorithms for reconfiguration variants 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 + ℓ . We also demonstrate the W[1] -hardness of reconfiguration variants of a large class of maximization problems parameterized by k + ℓ , and of their corresponding deletion problems parameterized by ℓ ; in doing so, we show that there exist problems in FPT when parameterized by k , but whose reconfiguration variants are W[1] -hard when parameterized by k + ℓ .