Result: Structural Parameterizations for Two Bounded Degree Problems Revisited

We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring , where the input is a graph G and a target degree Δ, and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by Δ. Both problems are known to be parameterized intractable for the most well-known structural parameters, such as treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. In particular, we present the following: — Both problems admit straightforward DP algorithms with table sizes \((\Delta +2)^\mathrm{tw}\) and \((\chi _\mathrm{d}(\Delta +1))^{\mathrm{tw}}\) , respectively, where tw is the input graph’s treewidth and \(\chi _\mathrm{d}\) the number of available colors. We show that, under the SETH, both algorithms are essentially optimal, for any non-trivial fixed values of \(\Delta , \chi _\mathrm{d}\) , even if we replace treewidth by pathwidth. Along the way, we obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for treewidth and pathwidth. — Given that the standard DP algorithm is optimal for treewidth and pathwidth, we then go on to consider the more restricted parameter tree-depth. Here, previously known lower bounds imply that, under the ETH, Bounded Vertex Degree Deletion and Defective Coloring cannot be solved in time \(n^{o(\sqrt [4]{\mathrm{td}})}\) and \(n^{o(\sqrt {\mathrm{td}})}\) , respectively, leaving some hope that a qualitatively faster algorithm than the one for treewidth may be possible. We close this gap by showing that neither problem can be solved in time \(n^{o(\mathrm{td})}\) , under the ETH, by employing a recursive low tree-depth construction that may be of independent interest. — Finally, we consider a structural parameter that is known to be restrictive enough to render both problems FPT: vertex cover. For both problems the best known algorithm in this setting has a super-exponential dependence of the form \(\mathrm{vc}^{\mathcal {O}(\mathrm{vc})}\) . We show that this is optimal, as an algorithm with dependence of the form \(\mathrm{vc}^{o(\mathrm{vc})}\) would violate the ETH. Our proof relies on a new application of the technique of d -detecting families introduced by Bonamy et al. [ToCT 2019]. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.