On the Complexity of Some Colorful Problems Parameterized by Treewidth

Fellows, Michael R.; Fomin, Fedor V.; Lokshtanov, Daniel; Rosamond, Frances A.; Saurabh, Saket; Szeider, Stefan; Thomassen, Carsten

doi:10.1007/978-3-540-73556-4_38

Cited by 24 publications

(27 citation statements)

References 22 publications

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“…Our results extend the recent results in [4,6]. These results studied the parameterized complexity of some NP-hard labeling and coloring problems with respect to graph-structural parameters, such as the treewidth, the vertex cover number, and the feedback vertex set number.…”

Section: Introductionsupporting

confidence: 82%

“…These results studied the parameterized complexity of some NP-hard labeling and coloring problems with respect to graph-structural parameters, such as the treewidth, the vertex cover number, and the feedback vertex set number. For example, it was shown in [4] that EC is W [1]-hard when parameterized by tw(G) and r combined. More recently, it was shown in [6] that EC is FPT when parameterized by vc(G).…”

Section: Introductionmentioning

confidence: 99%

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What Makes Equitable Connected Partition Easy

Enciso

Fellows

Guo

et al. 2009

Parameterized and Exact Computation

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Abstract. We study the Equitable Connected Partition problem, which is the problem of partitioning a graph into a given number of partitions, such that each partition induces a connected subgraph, and the partitions differ in size by at most one. We examine the problem from the parameterized complexity perspective with respect to the number of partitions, the treewidth, the pathwidth, the size of a minimum feedback vertex set, the size of a minimum vertex cover, and the maximum number of leaves in a spanning tree of the graph. In particular, we show that the problem is W[1]-hard with respect to the first four parameters (even combined), whereas it becomes fixed-parameter tractable when parameterized by the last two parameters. The hardness result remains true even for planar graphs. We also show that the problem is in XP when parameterized by the treewidth (and hence any other mentioned structural parameter). Furthermore, we show that the closely related problem, Equitable Coloring, is FPT when parameterized by the maximum number of leaves in a spanning tree of the graph.

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Section: Introductionsupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

What Makes Equitable Connected Partition Easy

Enciso

Fellows

Guo

et al. 2009

Parameterized and Exact Computation

Self Cite

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“…Using the above proof strategy, the result that k-List Coloring is W [1]-hard with respect to the parameter treewidth [10] can be transferred to IC k-List Coloring. Moreover, it also follows that IC k-List Coloring is NP-complete for fixed k ≥ 3 for all hereditary graph classes 1 where the ordinary List Coloring problem is NP-hard for fixed k ≥ 3, e. g., planar bipartite and chordal graphs.…”

Section: Parameterized Complexitymentioning

confidence: 99%

“…In order to show the W [1]-hardness, we present a parameterized reduction from the W [1]-complete k-Multicolored Independent Set problem [10,11]. The problem is to decide for a given k-coloring f for a graph G = (V, E) whether there exists a multicolored k-independent set, that is, a vertex subset…”

Section: Parameterized Complexitymentioning

confidence: 99%

“…In other words, the task is to extend a partial k-coloring to the entire graph. It has been shown that on general graphs PrExt is not fixed-parameter tractable with respect to parameter treewidth [10] but, other than List Coloring, it becomes fixed-parameter tractable when parameterized by vertex cover size [13]. Moreover, it is NP-complete for fixed k ≥ 3 on planar bipartite graphs [15] and W [1]-hard with respect to the number of precolored vertices for chordal graphs [16].…”

Section: Introductionmentioning

confidence: 99%

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Incremental List Coloring of Graphs, Parameterized by Conservation

Hartung

Niedermeier

2010

Lecture Notes in Computer Science

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Abstract. Incrementally k-list coloring a graph means that a graph is given by adding stepwise one vertex after another, and for each intermediate step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing some of in total k colors. We introduce the "conservative version" of this problem by adding a further parameter c ∈ specifying the maximum number of vertices to be recolored between two subsequent graphs (differing by one vertex). This "conservation parameter" c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing radical changes. We show that the problem is NP-hard for k ≥ 3 and W [1]-hard when parameterized by c. In contrast, the problem becomes fixed-parameter tractable with respect to the combined parameter (k, c). We prove that the problem has an exponential-size kernel with respect to (k, c) and there is no polynomial-size kernel unless NP ⊆ coNP/poly. Finally, we provide empirical findings for the practical relevance of our approach in terms of a very effective graph coloring heuristic.

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Iterated Type Partitions

Cordasco

Gargano

Rescigno

2020

Lecture Notes in Computer Science

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This paper introduces a novel parameter, called iterated type partition, that can be computed in polynomial time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parametrized by the iterated type partition. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parametrized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity. Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition, which enables us to find optimal solutions for several graph problems. While the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. As an example, in this paper, we give an algorithm for the Dominating Set problem that outputs an optimal set in time O(2 t + poly(n)), where n and t are the size and the iterated type partition of the input graph, respectively.

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On the Complexity of Some Colorful Problems Parameterized by Treewidth

Cited by 24 publications

References 22 publications

What Makes Equitable Connected Partition Easy

What Makes Equitable Connected Partition Easy

Incremental List Coloring of Graphs, Parameterized by Conservation

Iterated Type Partitions

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