Defective Coloring on Classes of Perfect Graphs

Belmonte, Rémy; Lampis, Michael; Mitsou, Valia

doi:10.46298/dmtcs.4926

Cited by 2 publications

(5 citation statements)

References 25 publications

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“…Conversely, algorithms which apply to treewidth apply also to all other parameters. matches asymptotically the exponent given in the algorithm of [9].…”

Section: Introductionsupporting

confidence: 63%

“…Second, can we determine the complexity of the problem with respect to other structural parameters, such as clique-width [15], modular-width [24], or neighborhood diversity [35]? For some of these parameters the existence of FPT algorithms is already ruled out by the fact that Defective Coloring is NP-hard on cographs [9], however the complexity of the problem is unknown if we also add χ d or ∆ * as a parameter. Finally, it would be very interesting to close the gap between 2 and 3/2 on the performance of the best treewidth-parameterized FPT approximation for χ d .…”

Section: Discussionmentioning

confidence: 99%

“…Theorem 19 gives a treewidth-based algorithm which can be obtained using standard techniques. Essentially the same algorithm was already sketched in [9], but we give another version here for the sake of completeness and because it is a building block for the approximation algorithm of Theorem 23. Theorem 20 uses a win/win argument to show that the problem is FPT parameterized by fvs when χ d = 2 and therefore explains why the reduction presented in Section 3 only works for 2 colors.…”

Section: Exact Algorithms For Treewidth and Other Parametersmentioning

confidence: 99%

“…We pose the natural question of whether similar success can be achieved for Defective Coloring, or whether the addition of ∆ * significantly alters the complexity behavior of the problem. Such results are not yet known for Defective Coloring, except for the fact that it was observed in [9] that the problem admits (by standard techniques) a roughly (χ d ∆ * ) tw -time algorithm, where tw is the graph's treewidth. In parameterized complexity terms, this shows that the problem is FPT parameterized by tw + ∆ * .…”

Section: Introductionmentioning

confidence: 99%

“…3,9,10,14,18,[20][21][22][23] have at most one endpoint outside {p A , p B }. Hence, by Lemmata 7, 10, we can conclude that td(H) = td(H \ {p A , p B }) + χ d + 1 and, for χ d = 2 we have fvs(H) ≤ fvs(H \ {p A , p B }) + 2, where H is the graph we obtain from H if we remove all the equality and palette gadgets.…”

mentioning

confidence: 99%

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Parameterized (Approximate) Defective Coloring

Belmonte¹,

Lampis²,

Mitsou³

2020

SIAM J. Discrete Math.

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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.

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“…Conversely, algorithms which apply to treewidth apply also to all other parameters. matches asymptotically the exponent given in the algorithm of [9].…”

Section: Introductionsupporting

confidence: 63%

Section: Discussionmentioning

confidence: 99%

Section: Exact Algorithms For Treewidth and Other Parametersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Parameterized (Approximate) Defective Coloring

Belmonte¹,

Lampis²,

Mitsou³

2020

SIAM J. Discrete Math.

Self Cite

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Structural Parameterizations for Two Bounded Degree Problems Revisited

Lampis,

Vasilakis

2024

ACM Trans. Comput. Theory

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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 problems are known to be parameterized intractable for the most well-known structural parameters, such as 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. In particular: • Both problems admit straightforward DP algorithms with table sizes ( Δ + 2) tw and ( χ d ( Δ + 1)) tw respectively, where tw is the input graph’s treewidth and χ d the number of available colors. We show that, under the SETH, both algorithms are essentially optimal, for any non-trivial fixed values of Δ , χ d , even if we replace treewidth by pathwidth. Along the way, we obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for treewidth and pathwidth. • Given that the standard DP algorithm is optimal for treewidth and pathwidth, we then go on to consider the more restricted parameter tree-depth. Here, previously known lower bounds imply that, under the ETH, Bounded Vertex Degree Deletion and Defective Coloring cannot be solved in time \(n^{o(\sqrt [4]{\mathrm{td}})} \) and \(n^{o(\sqrt {\mathrm{td}})} \) respectively, leaving some hope that a qualitatively faster algorithm than the one for treewidth may be possible. We close this gap by showing that neither problem can be solved in time n o (td) , under the ETH, by employing a recursive low tree-depth construction that may be of independent interest. • Finally, we consider a structural parameter that is known to be restrictive enough to render both problems FPT: vertex cover. For both problems the best known algorithm in this setting has a super-exponential dependence of the form \(\mathrm{vc}^{\mathcal {O}(\mathrm{vc})} \) . We show that this is optimal, as an algorithm with dependence of the form vc o (vc) would violate the ETH. Our proof relies on a new application of the technique of d -detecting families introduced by Bonamy et al. [ToCT 2019]. 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.

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Defective Coloring on Classes of Perfect Graphs

Cited by 2 publications

References 25 publications

Parameterized (Approximate) Defective Coloring

Parameterized (Approximate) Defective Coloring

Structural Parameterizations for Two Bounded Degree Problems Revisited

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