Parameterized (Approximate) Defective Coloring

Abstract

In Defective Coloring we are given a graph G = (V, E) and two integers χ d , ∆ * and are asked if we can partition V into χ d color classes, so that each class induces a graph of maximum degree ∆ * . 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 ETHbased 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 ∆ * , the problem admits an algorithm which for any > 0 runs in time (tw/ ) O(tw) and returns a solution with exactly the desired number of colors that approximates the optimal ∆ * within (1 + ). We also give a (tw) O(tw) algorithm which achieves the desired ∆ * 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.

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

“…The aim again is to minimize the number of used colors. In contrast to VERTEX COLORING, the DEFECTIVE COLORING problem is W[1]-hard [104] parameterized by the treewidth. This parameter measures how "tree-like" a graph is, and is defined as follows.…”

“…The strong polynomial-time approximation lower bound of n 1−ε for VERTEX COLORING [81] naturally carries over to the more general DEFECTIVE COLORING problem. A much improved approximation factor of 2 is possible though in FPT time if the parameter is the treewidth [104]. It can be shown however, that a PAS is not possible in this case, as there is no (3/2 − ε)-approximation algorithm for any ε > 0 parameterized by the treewidth [104], unless FPT = W[1].…”

“…A much improved approximation factor of 2 is possible though in FPT time if the parameter is the treewidth [104]. It can be shown however, that a PAS is not possible in this case, as there is no (3/2 − ε)-approximation algorithm for any ε > 0 parameterized by the treewidth [104], unless FPT = W[1]. A natural question is whether the bound ∆ of DEFECTIVE COLORING can be approximated instead of the number of colors.…”

“…A natural question is whether the bound ∆ of DEFECTIVE COLORING can be approximated instead of the number of colors. For this setting, a bicriteria PAS parameterized by the treewidth exists [104], which computes a solution with the optimum number of colors where each color class induces a graph of maximum degree at most (1 + ε)∆. The algorithms of the previous theorem build on the techniques of [105] using approximate addition trees in combination with dynamic programs that yield XP algorithms for these problems.…”

“…[104]). For the DEFECTIVE COLORING problem, given a tree decomposition of width k of the input graph,• a solution with the optimum number of colors where each color class induces a graph of maximum degree (1 + ε)∆ can be computed in (k/ε) O(k) n O(1) time for any ε > 0, • a 2-approximation (of the optimum number of colors) can be computed in k O(k) n O(1) time, but • no (3/2 − ε)-approximation (of the optimum number of colors) can be computed in f (k)n O(1) time for any ε > 0 and computable function f , unless FPT = W[1].…”