Parameterized Pre-Coloring Extension and List Coloring Problems

Gutin, Gregory; Majumdar, Diptapriyo; Ordyniak, Sebastian; Wahlström, Magnus

doi:10.1137/20m1323369

Cited by 9 publications

(11 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our second result is a (faster) randomized algorithm for DomCol parameterized by the size of a Clique Modulator, which is a subset of the vertex set whose deletion results in a clique. This algorithm uses an inclusion-exclusion based polynomial sieving technique introduced in a recent paper [23] in addition to an exact dynamic programming single-exponential algorithm to solve DomCol that we develop. We believe that our modification of the results in [23] is encouraging -can this powerful technique be used in other graph problems?…”

Section: Xx:4 Dominator Coloring Parameterized By Cluster Vertex Dele...mentioning

confidence: 99%

“…This algorithm uses an inclusion-exclusion based polynomial sieving technique introduced in a recent paper [23] in addition to an exact dynamic programming single-exponential algorithm to solve DomCol that we develop. We believe that our modification of the results in [23] is encouraging -can this powerful technique be used in other graph problems? Finally, we prove some loose lower bounds for DomCol with respect to these parameters.…”

Section: Xx:4 Dominator Coloring Parameterized By Cluster Vertex Dele...mentioning

confidence: 99%

“…Our algorithm follows a similar randomized strategy using polynomial sieving and Schwartz-Zippel Lemma, as that of [23] for list coloring parameterized by clique modulator. In [23], each v ∈ M is associated with a variable x v .…”

Section: Parameterized By Clique Modulator Sizementioning

confidence: 99%

See 2 more Smart Citations

Dominator Coloring Parameterized by Cluster Vertex Deletion Number

Banik¹,

Narasimhan²,

Raman³

2022

Preprint

View full text Add to dashboard Cite

The Dominator Coloring problem borrows the properties of two classical problems in graph theory -Graph Coloring and Dominating Set. A dominator coloring χ d of a graph G is a proper coloring of its vertices such that each vertex dominates a color class -that is, for each v ∈ V (G), there exists a color c such thatGiven a graph G and a natural number , the Dominator Coloring problem asks if there is a dominator coloring of G which uses at most -many colors. The problem, which was first described in 2006 and studied in several papers since then, still hosts several important open questions. While it is known that Dominator Coloring is FPT (Fixed-Parameter Tractable) when parameterized by (t, ) where is the number of colors used and t the treewidth of G, the structural parameterized landscape of the problem remains unexplored. We initiate the study of Dominator Coloring through the lens of structural parameterization.Our first result in this paper is a randomized O * (c k ) algorithm for the problem where c is some constant and k is the size of a graph's Clique Modulator, a set of vertices whose deletion results in a clique. This algorithm is obtained by a non-trivial adaptation of the recent work by Gutin et al. for List Coloring parameterized by the clique modulator that uses an inclusion-exclusion based polynomial sieving technique, and in addition uses a dynamic programming based exact algorithm we develop for Dominator Coloring.Later, we go on to prove the main result of the paper that Dominator Coloring is FPT when parameterized by the size of a graph's Cluster Vertex Deletion (CVD) set, in contrast to the W[i]-hardness result for List Coloring parameterized by the CVD set size. En route, we design a simpler and faster deterministic FPT algorithm when the problem is parameterized by the size of a graph's Twin Cover. We believe that this algorithm's approach, which uses a relationship between Dominator Coloring and List Coloring that we establish, is of independent interest.

show abstract

Section: Xx:4 Dominator Coloring Parameterized By Cluster Vertex Dele...mentioning

confidence: 99%

Section: Xx:4 Dominator Coloring Parameterized By Cluster Vertex Dele...mentioning

confidence: 99%

See 1 more Smart Citation

Dominator Coloring Parameterized by Cluster Vertex Deletion Number

Banik¹,

Narasimhan²,

Raman³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…For the standard (chromatic number) graph coloring, a linear sized kernel in k is known [4] for this parameterization using the combinatorial crown decomposition. Recently, a generalization of this for list coloring variants has also been shown to be fixed-parameter tractable [2,9] (See Sect. 2 for definitions on kernel and fixed-parameter tractability).…”

Section: R-role Coloringmentioning

confidence: 99%

Parameterizing Role Coloring on Forests

Pandey

Raman

Sahlot

2021

SOFSEM 2021: Theory and Practice of Computer Science

View full text Add to dashboard Cite

A role coloring of a graph is an assignment of colors to the vertices such that if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. In k-Role Coloring, we want to ask whether a given graph is role colorable by using exactly k colors.Determining whether a graph admits a k-Role Coloring is a notoriously hard problem even for a fixed k ≥ 2. It is known to be NPcomplete for k ≥ 2 on general graphs. For many hereditary graph classes like chordal graphs, planar graphs and split graphs, k-Role Coloring is NP-complete even when k is a constant. A recent result shows that k-Role Coloring is NP-complete for bipartite graphs when k ≥ 3. The only known classes of graphs for which k-role coloring is polynomial time for any fixed k, are Cographs and Proper Interval graphs.We consider the parameterized complexity of (n − k)-role coloring on n vertex graphs parameterized by k. This parameterization had interesting fixed-parameter tractable algorithms for the standard proper coloring [Chor et al. WG2004] and list-coloring variants [Banik et al. IWOCA 2019, Gutin et al. STACS 2020. As our main results, we show that this parameterization for role coloring has -an n O(k) algorithm on general graphs (putting the problem in the parameterized complexity class XP), and -an f (k)n O(1) algorithm (placing the problem in the class FPT) on forests (here f is a computable exponential function).

show abstract

“…A graph G is said to be k distance to clique if there exists a A ⊆ V (G) with |A| = k such that G[V \ A] is a clique. As shown in [19], for any graph G, determining whetheror not G is k distance to clique, can be computed in O * (1.2738 k ) time and can be approximated within a factor of two. Determining whether there is a vertex cover of size k, parameterized by k is FPT and can be computed in O * (1.2738 k ) time.…”

Section: Graph Notationsmentioning

confidence: 99%

Structural Parameterizations of Budgeted Graph Coloring

Bandopadhyay¹,

Banerjee²,

Banik³

et al. 2021

Preprint

View full text Add to dashboard Cite

We introduce a variant of the graph coloring problem, which we denote as Budgeted Coloring Problem (BCP). Given a graph G, an integer c and an ordered list of integers {b 1 , b 2 , . . . , b c }, BCP asks whether there exists a proper coloring of G where the i-th color is used to color at most b i many vertices. This problem generalizes two well-studied graph coloring problems, Bounded Coloring Problem (BoCP) and Equitable Coloring Problem (ECP) and as in the case of other coloring problems, it is NP-hard even for constant values of c. So we study BCP under the paradigm of parameterized complexity, particularly with respect to (structural) parameters that specify how far (the deletion distance) the input graph is from a tractable graph class.• We show that BCP is FPT (fixed-parameter tractable) parameterized by the vertex cover size. This generalizes a similar result for ECP and immediately extends to the BoCP, which was earlier not known. • We show that BCP is polynomial time solvable for cluster graphs generalizing a similar result for ECP. However, we show that BCP is FPT, but unlikely to have polynomial kernel, when parameterized by the deletion distance to clique, contrasting the linear kernel for ECP for the same parameter. • While the BoCP is known to be polynomial time solvable on split graphs, we show that BCP is NP-hard on split graphs. As BoCP is hard on bipartite graphs when c > 3, the result follows for BCP as well. We provide a dichotomy result by showing that BCP is polynomial time solvable on bipartite graphs when c = 2. We also show that BCP is NP-hard on co-cluster graphs, contrasting the polynomial time algorithm for ECP and BoCP. Finally we present an O * (2 |V (G)| ) algorithm for the BCP, generalizing the known algorithm with a similar bound for the standard chromatic number.

show abstract

Parameterized Pre-Coloring Extension and List Coloring Problems

Cited by 9 publications

References 27 publications

Dominator Coloring Parameterized by Cluster Vertex Deletion Number

Dominator Coloring Parameterized by Cluster Vertex Deletion Number

Parameterizing Role Coloring on Forests

Structural Parameterizations of Budgeted Graph Coloring

Contact Info

Product

Resources

About