On the computational complexity of (O,P)-partition problems

Kratochvı́l, Jan; Schiermeyer, Ingo

doi:10.7151/dmgt.1052

Cited by 12 publications

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“…To this end, we consider the first level above triviality of the problem. When Π A is characterized by a forbidden induced subgraph of order 2, then (Π A , Π B )-Recognition can be solved in linear time [18], and thus we focus on the NP-hard case when the forbidden induced subgraph has order 3 [2,14,25]. In particular, we let Π A be the class of so-called cluster graphs.…”

Section: Our Resultsmentioning

confidence: 99%

“…Cluster-Π-Partition generalizes the recognition problem of many graph classes, such as the recognition of monopolar graphs [6,8,9,26] (Π is the set of edgeless graphs), 2-subcolorable graphs [5,16,20,29] (Π is the set of cluster graphs), and several others [1,4,7]. Unfortunately, Cluster-Π-Partition is NP-hard in these special cases, and in general when Π is characterized by a set of connected forbidden induced subgraphs [2,14,25]. Hence, we consider the number k of clusters in the cluster graph G[A] as a parameter, and study the pushing process with respect to this parameter.…”

Section: Our Resultsmentioning

confidence: 99%

“…We call (A, B) a (Π A , Π B )-partition of G. The (Π A , Π B )-Recognition problem is to recognize whether a given graph is a (Π A , Π B )-graph. This captures a wealth of famous problems, including the recognition of 3-colorable, bipartite, co-bipartite, and split graphs, and Π-Vertex Deletion, which asks for a partition (A, B) such that G[A] ∈ Π and G [B] has order at most k for some given k. 1 In the most interesting (and NP-hard) cases [2,14,25], Π A and Π B are both characterized by a (not necessarily finite) set of forbidden connected 2 induced subgraphs. In other words, Π A and Π B are each closed under the disjoint union of graphs in these cases.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Kanj¹,

Komusiewicz²,

Sorge³

et al. 2020

SIAM J. Discrete Math.

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A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G [A] and G [B] satisfy properties ΠA and ΠB, respectively. This so-called (ΠA, ΠB)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (ΠA, ΠB)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A , B ), and pushes appropriate vertices from A to B and vice versa to eventually arrive at a correct bipartition.In this paper, we study whether (ΠA, ΠB)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where ΠA is the set of P3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and ΠB is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP ⊆ coNP/poly, (ΠA, ΠB)-Recognition admits a polynomial kernel if and only if H contains a graph with at most 2 vertices. In both the kernelization and the lower bound results, we make crucial use of the pushing process. Theorem 1.3. Cluster-Π ∆ -Partition parameterized by k, the maximum size of B, has an O((∆ 2 + 1) · k 2 )-vertex kernel. PreliminariesGraphs We follow standard graph-theoretic notation [11]. Let G be a graph. By V (G) and E(G) we denote the vertex-set and the edge-set of G, respectively. Throughout the paper, we use n := |V (G)| to denote the number of vertices in G and m := |E(G)| to denote its number of edges. We also say that G is of order |V (G)|. We assume n = O(m) since isolated vertices can be safely removed in the problems that we consider. For X ⊆ V (G), G[X] = (X, {e | e ∈ E(G) ∩ X} denotes the subgraph of G induced by X. For a vertex v ∈ G, N (v) = {u | {u, v} ∈ E(G)} and N [v] = N (v) ∪ {v} denote the open neighborhood and the closed neighborhood of v, respectively. For X ⊆ V (G), we define N (X) := ( v∈X N (v)) \ X and N [X] := v∈X N [v], and for a family X of subsets X ⊆ V (G), we define N (X ) := ( X∈X N (X)) \ ( X∈X X) and N [X ] := X∈X N [X].

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Kanj¹,

Komusiewicz²,

Sorge³

et al. 2020

SIAM J. Discrete Math.

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show abstract

“…Second, if P is an additive hereditary class other than the class of empty graphs, then (O, P)-PARTI-TION is NP-complete [7]. In particular, the problem is NP-complete if P = P 2 .…”

Section: Theoremmentioning

confidence: 99%

Between 2- and 3-colorability

Lozin

2005

Information Processing Letters

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“…We say that Π is a problem with compositive solutions if such an algorithm A 2 exists (note that such algorithms can be easily given for various domination and coloring problems). A major drawback of such a three-phase approach (partition, solve, compose) is that we do not know any simple realization of this procedure, because, with a sole exception of proper 2-Coloring, the P-coloring problem is computationally hard (see Brown 1996;Kratochvíl and Schiermeyer 1997). Furthermore, our knowledge of minimal forbidden subgraphs that characterize C(P, k) is far from being complete (for some recent results in context of proper k-coloring of graphs in selected classes, mainly for k ∈ {3, 4}, see, e.g., Hoàng et al (2015), Chudnovsky et al (2016), and Goedgebeur and Schaudt (2016)).…”

Section: Motivation Critical Partitions and Minimal Graphsmentioning

confidence: 99%

Computational aspects of greedy partitioning of graphs

Borowiecki

2017

J Comb Optim

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In this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related too greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs P-coloring with at least k colors is NPcomplete if P is a class of K p -free graphs with p ≥ 3. On the other hand we give a polynomial-time algorithm when k is fixed and the family of minimal forbidden graphs defining the class P is finite. We also prove coNP-completeness of deciding if for a given graph G and an integer t ≥ 0 the difference between the largest number of colors used by the greedy algorithm and the minimum number of colors required in any P-coloring of G is bounded by t. In view of computational hardness, we present new Brooks-type bound on the largest number of colors used by the greedy P-coloring algorithm.

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On the computational complexity of (O,P)-partition problems

Abstract: We prove that for any additive hereditary property P > O, it is NP-hard to decide if a given graph G allows a vertex partition V (G) = A ∪ B such that G[A] ∈ O (i.e., A is independent) and G[B] ∈ P.

Cited by 12 publications

References 1 publication

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

Between 2- and 3-colorability

Computational aspects of greedy partitioning of graphs

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