Parameterized Complexities of Dominating and Independent Set Reconfiguration

TL;DR

This paper determines the parameterized computational complexities of various independent set and dominating set reconfiguration problems, establishing their completeness in several complexity classes and exploring equivalences between different reconfiguration models.

Contribution

It provides a comprehensive complexity classification for these reconfiguration problems and introduces new variants with precise complexity characterizations.

Findings

Problems are XL-, XNL-, and XNLP-complete depending on move limits.

Token sliding and token jumping are equivalent under pl-reductions.

Introduces partitioned variants with controlled parameters and reductions.

Abstract

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves, XNL-complete when a maximum length $\ell$ for the sequence is given in binary in the input, and XNLP-complete when $\ell$ is given in unary. The problems were known to be $\mathrm{W}[1]$- and $\mathrm{W}[2]$-hard respectively when $\ell$ is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.