Fixed-parameter tractability of graph modification problems for hereditary properties

Cai, Lilong

doi:10.1016/0020-0190(96)00050-6

Cited by 359 publications

(302 citation statements)

References 7 publications

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“…Cluster Editing, parameterized by the number k of false positives and negatives, is FPT [3,9,10]. Furthermore, it has a polynomial-time 4-approximation algorithm but it does not admit a PTAS unless P = NP [14].…”

Section: Introductionmentioning

confidence: 99%

Generalized Graph Clustering: Recognizing (p,q)-Cluster Graphs

Heggernes

Lokshtanov

Nederlof

et al. 2010

Lecture Notes in Computer Science

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Cluster Editing is a classical graph theoretic approach to tackle the problem of data set clustering: it consists of modifying a similarity graph into a disjoint union of cliques, i.e, clusters. As pointed out in a number of recent papers, the cluster editing model is too rigid to capture common features of real data sets. Several generalizations have thereby been proposed. In this paper, we introduce (p, q)-cluster graphs, where each cluster misses at most p edges to be a clique, and there are at most q edges between a cluster and other clusters. Our generalization is the first one that allows a large number of false positives and negatives in total, while bounding the number of these locally for each cluster by p and q. We show that recognizing (p, q)-cluster graphs is NP-complete when p and q are input. On the positive side, we show that (0, q)-cluster, (p, 1)-cluster, (p, 2)-cluster, and (1, 3)-cluster graphs can be recognized in polynomial time.

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Section: Introductionmentioning

confidence: 99%

Generalized Graph Clustering: Recognizing (p,q)-Cluster Graphs

Heggernes

Lokshtanov

Nederlof

et al. 2010

Lecture Notes in Computer Science

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“…For the parametric dual problem of Induced (Π b , Π r )-Subgraph, i.e., deleting k vertices to obtain an induced (Π b , Π r )-graph, we can easily deduce the following general result on edge-colored multigraphs as a corollary of a well-known result of the first author regarding graph modification problems [3]. …”

Section: Theorem 5 Dual Separator Is Np-completementioning

confidence: 97%

Dual Connectedness of Edge-Bicolored Graphs and Beyond

Cai

2014

Mathematical Foundations of Computer Science 2014

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Abstract. Let G be an edge-bicolored graph where each edge is colored either red or blue. We study problems of obtaining an induced subgraph H from G that simultaneously satisfies given properties for H's red graph and blue graph. In particular, we consider Dually Connected Induced Subgraph problem -find from G a k-vertex induced subgraph whose red and blue graphs are both connected, and Dual Separator problem -delete at most k vertices to simultaneously disconnect red and blue graphs of G.We will discuss various algorithmic and complexity issues for Dually Connected Induced Subgraph and Dual Separator problems: NP-completeness, polynomial-time algorithms, W[1]-hardness, and FPT algorithms. As by-products, we deduce that it is NP-complete and W[1]-hard to find k-vertex (resp., (n − k)-vertex) strongly connected induced subgraphs from n-vertex digraphs. We will also give a complete characterization of the complexity of the problem of obtaining a k-vertex induced subgraph H from G that simultaneously satisfies given hereditary properties for H's red and blue graphs.

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“…Cai [7] proved that for any property defined by a finite set of forbidden induced subgraphs, the editing problem is FPT when parameterized by the bound on the number of vertex deletions, edge deletions and edge additions. Further results for hereditary properties were obtained by Khot and Raman [15].…”

Section: Introductionmentioning

confidence: 99%

Editing to a Graph of Given Degrees

Golovach

2014

Parameterized and Exact Computation

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We consider the Editing to a Graph of Given Degrees problem that asks for a graph G, non-negative integers d, k and a function δ : V (G) → {1, . . . , d}, whether it is possible to obtain a graph G ′ from G such that the degree of v is δ(v) for any vertex v by at most k vertex or edge deletions or edge additions. We construct an FPT-algorithm for Editing to a Graph of Given Degrees parameterized by d+k. We complement this result by showing that the problem has no polynomial kernel unless NP ⊆ coNP /poly.

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Fixed-parameter tractability of graph modification problems for hereditary properties

Cited by 359 publications

References 7 publications

Generalized Graph Clustering: Recognizing (p,q)-Cluster Graphs

Generalized Graph Clustering: Recognizing (p,q)-Cluster Graphs

Dual Connectedness of Edge-Bicolored Graphs and Beyond

Editing to a Graph of Given Degrees

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