Parameterized (Approximate) Defective Coloring
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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 > 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.
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