Parameterized (Approximate) Defective Coloring

In Defective Coloring we are given a graph $G = (V, E)$ and two integers $χ_d, Δ^*$ and are asked if we can partition $V$ into $χ_d$ color classes, so that each class induces a graph of maximum degree $Δ^*$. 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 $χ_d = 2$. As expected, this hardness can be extended to larger values of $χ_d$ for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any $χ_d \ge 2$, and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETH-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 $Δ^*$, the problem admits an algorithm which for any $ε> 0$ runs in time $(tw/ε)^{O(tw)}$ and returns a solution with exactly the desired number of colors that approximates the optimal $Δ^*$ within $(1 + ε)$. We also give a $(tw)^{O(tw)}$ algorithm which achieves the desired $Δ^*$ exactly while 2-approximating the minimum value of $χ_d$. 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 $χ_d$, even when an extra constant additive error is also allowed.