## Complexity classification of some edge modification problems (2001)

Citations: | 41 - 2 self |

### BibTeX

@MISC{Natanzon01complexityclassification,

author = {Assaf Natanzon and Ron Shamir and Roded Sharan},

title = {Complexity classification of some edge modification problems },

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.

### Citations

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Citation Context ... other vertex of G. For a graph property \Pi the notation G 2 \Pi implies that G satisfied \Pi . For basic definitions of graph properties and much more on the graph classes discussed here see, e.g., =-=[13, 6]-=-. Let \Pi be a graph property. If F is a set of non-edgessuch that G 0 = (V; E[F ) 2 \Pi and jF jsk, then F is called a k-completion set w.r.t. \Pi . k-deletion set and k-editing set are similarly def... |

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Citation Context ...ror. We summarize below some of these applications. Definitions of the graph classes are given in Section 3. Interval modification problems have important applications in physical mapping of DNA (see =-=[5, 8, 12, 14]-=-). Depending on the biotechnology used and the kind of experimental errors, completion, deletion and editing problem arise, both for interval graphs and for unit interval graphs. The chordal completio... |

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Citation Context ...idden edges. Polynomial algorithms or NP-hardness results are known for many sandwich problems [16, 15, 18, 21]. Several results on the parametric complexity of completion problems were also obtained =-=[22, 7]-=-. Approximation algorithms exist for several problems. In [28] an 8k approximation algorithm is given for the minimum fill-in problem, where k denotes the size of an optimum solution. In [1] an O(m 1=... |

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Citation Context ...ror. We summarize below some of these applications. Definitions of the graph classes are given in Section 3. Interval modification problems have important applications in physical mapping of DNA (see =-=[5, 8, 12, 14]-=-). Depending on the biotechnology used and the kind of experimental errors, completion, deletion and editing problem arise, both for interval graphs and for unit interval graphs. The chordal completio... |

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Citation Context ...ained [22, 7]. Approximation algorithms exist for several problems. In [28] an 8k approximation algorithm is given for the minimum fill-in problem, where k denotes the size of an optimum solution. In =-=[1]-=- an O(m 1=4 log 3:5 n) approximation algorithm is given for the minimum chordal supergraph problem (where one wishes to minimize the total number of edges in the resulting graph) For the minimum inter... |

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Citation Context ...wn for many sandwich problems [16, 15, 18, 21]. Several results on the parametric complexity of completion problems were also obtained [22, 7]. Approximation algorithms exist for several problems. In =-=[28]-=- an 8k approximation algorithm is given for the minimum fill-in problem, where k denotes the size of an optimum solution. In [1] an O(m 1=4 log 3:5 n) approximation algorithm is given for the minimum ... |

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Citation Context ...ussian elimination on a sparse symmetric positive-definite matrix [30]. Chordal deletion problems occur when trying to solve the CLIQUE problem. Some heuristics for finding a large clique (see, e.g., =-=[33]-=-) aim to find a maximum chordal subgraph of the input graph, on which a maximum clique can be found in polynomial time. Previous Results: Strong negative results are known for vertex deletion problems... |

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Citation Context ...andwich problems, seek a supergraph satisfying a given property which does not include (pre-defined) forbidden edges. Polynomial algorithms or NP-hardness results are known for many sandwich problems =-=[16, 15, 18, 21]-=-. Several results on the parametric complexity of completion problems were also obtained [22, 7]. Approximation algorithms exist for several problems. In [28] an 8k approximation algorithm is given fo... |

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Citation Context ...ordal supergraph problem (where one wishes to minimize the total number of edges in the resulting graph) For the minimum interval supergraph problem an O(log 2 n) approximation algorithm was given in =-=[29]-=-. In [8] it was shown that the minimum number of edge editions needed in order to convert a graph into a caterpillar cannot be approximated in polynomial time to within an additive term of O(n 1\Gamma... |

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Citation Context ...on and Cograph Deletion were shown to be NP-complete in [10]. Threshold Completion and Threshold Deletion were shown to be NP-complete in [27]. Comparability Completion was shown to be NP-complete in =-=[17]-=- and Comparability Deletion was shown to be NP-complete in [35]. Much fewer results are known for editing problems: Chordal Editing was proven to be NP-complete in [4]. The connected bipartite interva... |

10 |
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Citation Context ...hreshold Completion and Threshold Deletion were shown to be NP-complete in [27]. Comparability Completion was shown to be NP-complete in [17] and Comparability Deletion was shown to be NP-complete in =-=[35]-=-. Much fewer results are known for editing problems: Chordal Editing was proven to be NP-complete in [4]. The connected bipartite interval (caterpillar) editing problem was proven to be NP-complete in... |

9 |
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Citation Context ...number of vertices and edges, respectively, in a graph). For edge modification problems no such general results are known, although some attempts have been made to go beyond specific graph properties =-=[3, 10, 2]-=-. Most of the results obtained so far concerning edge modification problems are NP-hardness ones. (For simplicity we shall often refer to the decision version of the optimization problems). Chain Comp... |

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Citation Context ...number of vertices and edges, respectively, in a graph). For edge modification problems no such general results are known, although some attempts have been made to go beyond specific graph properties =-=[3, 10, 2]-=-. Most of the results obtained so far concerning edge modification problems are NP-hardness ones. (For simplicity we shall often refer to the decision version of the optimization problems). Chain Comp... |

7 |
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Citation Context ...graphs were proven to be NP-complete in [12]. Cograph Completion and Cograph Deletion were shown to be NP-complete in [10]. Threshold Completion and Threshold Deletion were shown to be NP-complete in =-=[27]-=-. Comparability Completion was shown to be NP-complete in [17] and Comparability Deletion was shown to be NP-complete in [35]. Much fewer results are known for editing problems: Chordal Editing was pr... |

6 |
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Citation Context ...number of vertices and edges, respectively, in a graph). For edge modification problems no such general results are known, although some attempts have been made to go beyond specific graph properties =-=[3, 10, 2]-=-. Most of the results obtained so far concerning edge modification problems are NP-hardness ones. (For simplicity we shall often refer to the decision version of the optimization problems). Chain Comp... |

5 | Bounded degree interval sandwich problems, Algorithmica 24 - Kaplan, Shamir - 1999 |

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(Show Context)
Citation Context ...ror. We summarize below some of these applications. Definitions of the graph classes are given in Section 3. Interval modification problems have important applications in physical mapping of DNA (see =-=[5, 8, 12, 14]-=-). Depending on the biotechnology used and the kind of experimental errors, completion, deletion and editing problem arise, both for interval graphs and for unit interval graphs. The chordal completio... |

4 |
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Citation Context |

3 |
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Citation Context ...free, or AT-free, if G contains no asteroidal triple. Several families of graphs are asteroidal triple free, e.g., interval and cocomparability graphs. For characterizations of AT-free graphs see cf. =-=[9]-=-. We prove here that AT-free Deletion is NP-complete. Theorem 5. AT-free Deletion is NP-complete. Proof. The problem is clearly in NP. The hardness proof is by reduction from Chain Deletion. Let ! G =... |

3 | Physical maps and interval sandwich problems: Bounded degrees help
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Citation Context |

3 | Fixed-parameter tractability of graph modi cation problems for hereditary properties - Cai - 1996 |

3 | Four strikes against physical mapping of - Goldberg, Golumbic, et al. - 1995 |

2 | Complexity classiffication of some edge modification problems - Natazon, Shamir, et al. |

2 | de Fluiter, On intervalizing k-colored graphs for DNA physical mapping - Bodlaender, B - 1996 |

1 |
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Citation Context ... other vertex of G. For a graph property \Pi the notation G 2 \Pi implies that G satisfied \Pi . For basic definitions of graph properties and much more on the graph classes discussed here see, e.g., =-=[13, 6]-=-. Let \Pi be a graph property. If F is a set of non-edgessuch that G 0 = (V; E[F ) 2 \Pi and jF jsk, then F is called a k-completion set w.r.t. \Pi . k-deletion set and k-editing set are similarly def... |

1 |
A characterization of perfect graphs
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(Show Context)
Citation Context ...f length 5. Otherwise, fp1 ; q1 ,v 1 q 1 ;q 2 ;i ,v 2 q 1 ;q 2 ;i ,v 3 q 1 ;q 2 ;i , q2 ; p2g induce a cycle of length 7. In any case we arrive at a contradiction. The perfect graph theorem by Lovasz =-=[25]-=- states that the complement of a perfect graph is perfect. Hence, we conclude that Perfect Deletion is also NP-hard. Theorem 2. Perfect Editing is NP-hard. Proof. Reduction from Chain Completion. Let ... |

1 | A characterization of perfect graphs - AS - 1972 |