Parameterized (Approximate) Defective Coloring
Rémy Belmonte, Michael Lampis, Valia Mitsou · SIAM Journal on Discrete Mathematics · 2020
In Defective Coloring we are given a graph $G=(V,E)$ and two integers ${\chi_d},\Delta^*$ and are asked if we can partition $V$ into ${\chi_d}$ color classes, so that each class induces a graph of maximum degree $\Delta^*$. We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set if ${\chi_d}=2$. As expected, this hardness can be extended to larger values of ${\chi_d}$ for most of these parameters, with one surprising exception: we show that the problem is fixed parameter tractable (FPT) and parameterized by feedback vertex set for any ${\chi_d} eq 2$, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an exponential time hypothesis-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the problem in $n^{o({{pw}})}$, essentially matching the complexity of an algorithm obtained with standard techniques. We complement these results by considering the problem's approximability and show that, with respect to $\Delta^*$, the problem admits an algorithm which for any $\epsilon>0$ runs in time $({{tw}}/\epsilon)^{O({{tw}})}$ and returns a solution with exactly the desired number of colors that approximates the optimal $\Delta^*$ within $(1+\epsilon)$. We also give a $({{tw}})^{O({{tw}})}$ algorithm which achieves the desired $\Delta^*$ exactly while 2-approximating the minimum value of ${\chi_d}$. We show that this is close to optimal, by establishing that no FPT algorithm can (under standard assumptions) achieve a better than 3/2-approximation to ${\chi_d}$, even when an extra constant additive error is also allowed.