NP-completeness results for edge modification problems

Burzyn, Pablo; Bonomo, Flavia; Durán, Guillermo

doi:10.1016/j.dam.2006.03.031

Cited by 76 publications

(50 citation statements)

References 25 publications

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“…It is easy to see that a flow of value 1 2 v∈V | bal(v)| in this network corresponds to an Eulerian extension for G and, thus, the minimum cost of such a flow is also the minimum cost of an Eulerian extension for G. Such a flow can be computed in O(m 2 + nm log n)) time. 4 ⊓ ⊔ Next, for a directed multigraph M let G M be the complete digraph (containing all possible arcs) on the vertex set of M . Analogously to the proof of Proposition 2, we can use a min-cost flow algorithm on G M with arc capacities ∞ and weights according to ω to solve WEE on connected directed multigraphs M .…”

Section: Polynomial-time Cases Of Eulerian Extensionmentioning

confidence: 99%

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Efficient Algorithms for Eulerian Extension

Dorn

Moser

Niedermeier

et al. 2010

Graph Theoretic Concepts in Computer Science

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Abstract. Eulerian extension problems aim at making a given (directed) (multi-)graph Eulerian by adding a minimum-cost set of edges (arcs). These problems have natural applications in scheduling and routing and are closely related to the Chinese Postman and Rural Postman problems. Our main result is to show that the NP-hard Weighted Multigraph Eulerian Extension is fixed-parameter tractable with respect to the number k of extension edges (arcs). For an n-vertex multigraph, the corresponding running time amounts to O(4 k · n 3 ). This implies a fixed-parameter tractability result for the "equivalent" Rural Postman problem. In addition, we present several polynomial-time algorithms for natural Eulerian extension problems.

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Section: Polynomial-time Cases Of Eulerian Extensionmentioning

confidence: 99%

“…Edge modification problems in graphs have many applications and are wellstudied in algorithmic graph theory [4,14]. The corresponding minimization problems ask to modify as few (potential) edges as possible such that an input graph is transformed into a graph with a desired property.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Eulerian Extension

Dorn

Moser

Niedermeier

et al. 2010

Graph Theoretic Concepts in Computer Science

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“…The edge deletion problems were considered by Yannakakis [23], Alon, Shapira and Sudakov [1]. The case when edge additions and deletions are allowed and the property is the inclusion in some hereditary graph class was considered by Natanzon, Shamir and Sharan [20] and Burzyn, Bonomo and Durán [6].…”

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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“…However this fact does not imply the existence of a polynomial kernelization. Related Work: There exists rich literature on the complexity of edge modification problems; recent surveys were given by Natanzon et al [17] and Burzyn et al [6]. Cai [7] showed that a very general version of this problem, also allowing vertex deletions, is FPT when the desired property Π can be characterized by a finite set of forbidden induced subgraphs.…”

Section: Introductionmentioning

confidence: 99%

“…Edge modification problems have a number of applications, including machine learning, numerical algebra, and molecular biology [6,[17][18][19]. In typical applications the input graphs arise from experiments and edge modification serves to correct the (hopefully) few errors.…”

Section: Introductionmentioning

confidence: 99%

Two Edge Modification Problems without Polynomial Kernels

Kratsch

Wahlström

2009

Parameterized and Exact Computation

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Abstract. Given a graph G and an integer k, the Π Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property Π. Edge modification problems have received considerable interest from a parameterized point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tractable and some are known to admit polynomial kernelizations, i.e. efficient preprocessing with a size guarantee that is polynomial in k. This paper answers an open problem posed by Cai (IWPEC 2006), namely, whether the Π Edge Deletion problem, parameterized by the number of deletions, admits a polynomial kernelization when Π can be characterized by a finite set of forbidden induced subgraphs. We answer this question negatively based on recent work by Bodlaender et al. (ICALP 2008) which provided a framework for proving polynomial lower bounds for kernelizability. We present a graph H on seven vertices such that H-free Edge Deletion and H-free Edge Editing do not admit polynomial kernelizations, unless NP ⊆ coNP/poly. The application of the framework is not immediate and requires a lower bound for a Not-1-in-3 SAT problem that may be of independent interest.

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NP-completeness results for edge modification problems

Cited by 76 publications

References 25 publications

Efficient Algorithms for Eulerian Extension

Efficient Algorithms for Eulerian Extension

Editing to a Graph of Given Degrees

Two Edge Modification Problems without Polynomial Kernels

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