Structural Parameterizations for Two Bounded Degree Problems Revisited
Michael Lampis, Manolis Vasilakis
TL;DR
The paper investigates two natural graph problems, Bounded Degree Vertex Deletion and Defective Coloring, under structural parameterizations such as treewidth, pathwidth, tree-depth, and vertex cover. It shows that the standard dynamic programming approaches with table sizes $ (\Delta+2)^{\mathrm{tw}} $ for Bounded Degree Vertex Deletion and $ (\chi_{\mathrm{d}}(\Delta+1))^{\mathrm{tw}} $ for Defective Coloring are tight under SETH, even when parameterizing by pathwidth, and provides a quasi-linear time Defective Coloring DP via FFT. For more restrictive parameters, it proves ETH-based lower bounds (e.g., no $n^{o(\mathrm{td})}$-time algorithms under ETH for tree-depth) and shows ETH-based lower bounds for vertex-cover parameterization using the technique of $d$-detecting families, complemented by a recursive low tree-depth construction. Collectively, the results yield a precise landscape of the parameterized complexity of these problems, demonstrating matching upper and lower bounds across several regimes and clarifying the limits of the treewidth toolbox.
Abstract
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 are known to be parameterized intractable for 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. Both admit straightforward DP algorithms with table sizes $(Δ+2)^\mathrm{tw}$ and $(χ_\mathrm{d}(Δ+1))^{\mathrm{tw}}$ respectively, where tw is the input graph's treewidth and $χ_\mathrm{d}$ the number of available colors. We show that both algorithms are optimal under SETH, even if we replace treewidth by pathwidth. Along the way, we also obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for these parameters. We then consider the more restricted parameter tree-depth, and bridge the gap left by known lower bounds, by showing that neither problem can be solved in time $n^{o(\mathrm{td})}$ under ETH. In order to do so, we employ a recursive low tree-depth construction that may be of independent interest. Finally, we show that for both problems, an $\mathrm{vc}^{o(\mathrm{vc})}$ algorithm would violate ETH, thus already known algorithms are optimal. Our proof relies on a new application of the technique of $d$-detecting families introduced by Bonamy et al. 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.