## Optimal Edge Deletions for Signed Graph Balancing (2007)

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Citations: | 9 - 4 self |

### BibTeX

@MISC{Hüffner07optimaledge,

author = {Falk Hüffner and Nadja Betzler and Rolf Niedermeier},

title = {Optimal Edge Deletions for Signed Graph Balancing },

year = {2007}

}

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### Abstract

The Balanced Subgraph problem (edge deletion variant) asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. We present an exact algorithm for Balanced Subgraph based on a combination of data reduction rules and a fixed-parameter algorithm. The data reduction is based on finding small separators and a novel gadget construction scheme. The fixed-parameter algorithm is based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological realworld instances exactly for which previously only approximations [DasGupta et al., WEA 2006] were known.

### Citations

904 |
Approximation algorithms
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- 2004
(Show Context)
Citation Context ...h problem is to find a balanced subgraph with maximum number of edges. Balanced Subgraph is a generalization of the NPhard Maximum Cut problem in graphs. Based on semidefinite programming for Max2SAT =-=[20]-=-, DasGupta et al. [3] developed an approximation algorithm that guarantees a solution with at least 87.9% of the number of edges of an optimal solution. They further showed that this approximation fac... |

873 | Parameterized Complexity - Downey, Fellows - 1999 |

365 |
Parameterized Complexity Theory
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- 2006
(Show Context)
Citation Context ...applications using a polynomial-time approximation algorithm of the minimization version seems of hardly any help. By way of contrast, we have good news from the field of fixed-parameter algorithmics =-=[4, 6,16]-=-. A problem is called fixed-parameter tractable with respect to a parameter k if an instance of size n can be solved in f(k) · n O(1) time, where f is an arbitrary function depending only on k. It is ... |

286 | Invitation to Fixed-Parameter Algorithms
- Niedermeier
- 2006
(Show Context)
Citation Context ...applications using a polynomial-time approximation algorithm of the minimization version seems of hardly any help. By way of contrast, we have good news from the field of fixed-parameter algorithmics =-=[4, 6,16]-=-. A problem is called fixed-parameter tractable with respect to a parameter k if an instance of size n can be solved in f(k) · n O(1) time, where f is an arbitrary function depending only on k. It is ... |

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- 2002
(Show Context)
Citation Context ... algorithm finds in polynomial time a solution of size O(k log k), where k is the size of an optimal solution [1]. It has been conjectured that it is NP-hard to improve the log k-factor to a constant =-=[15]-=-. Hence, with respect to practical applications using a polynomial-time approximation algorithm of the minimization version seems of hardly any help. By way of contrast, we have good news from the fie... |

171 |
Dividing A Graph into Triconnected Components
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- 1973
(Show Context)
Citation Context ...y flipping all colors in one component), since they overlap only in one vertex. Note that this phase in particular removes all degree-1 vertices. Separators of size 2 can also be found in linear time =-=[13]-=-. However, we did not implement this algorithm, since it is quite complicated and error-prone to implement (several errors in the original publication have been pointed out [10]). Separators of size k... |

91 |
On the notion of balance in a signed graph
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Citation Context ...ogical realworld instances exactly for which previously only approximations [DasGupta et al., WEA 2006] were known. 1 Introduction The concept of balanced signed graphs was first introduced by Harary =-=[11]-=- for the analysis of social networks (see [22] for a bibliography of signed graphs), and has been frequently rediscovered since. In particular, motivated by the mathematical analysis of large-scale bi... |

83 |
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(Show Context)
Citation Context ...mial time an instance to a smaller instance, without destroying the possibility of finding an optimal solution. Data reduction has proven useful as a general technique in coping with NP-hard problems =-=[9]-=-. The corresponding reduction rules, however, have to be developed in a problem-specific way. In this section, we describe an effective data reduction scheme for the Balanced Subgraph problem. The sch... |

36 |
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Citation Context ... end the section of regulatory networks, we describe our results for two larger networks that yet cannot be solved optimally with our method. Regarding the regulatory network for a toll-like receptor =-=[18]-=-, we could reduce the number of vertices from 688 to 244 and the number of edges from 2208 to 1159 within three minutes. For the regulatory network of the archaeon Methanosarcina barkeri [5], we were ... |

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- 2001
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Citation Context ...e found in linear time [13]. However, we did not implement this algorithm, since it is quite complicated and error-prone to implement (several errors in the original publication have been pointed out =-=[10]-=-). Separators of size k for small k can be found efficiently by flow techniques [12]. However, after some experiments we settled for the subsequently described heuristic instead, which is faster and p... |

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- 2000
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Citation Context ...arators of size 0 (that is, the graph consists of more than one connected component) and separators of size 1 (that is, articulation points). They can be found in linear time using depth-first search =-=[7]-=-. For these cases, the gadget construction can be omitted: the 2-connected components 1 can be treated independently, and optimal colorings of two components can always be merged (possibly by flipping... |

27 |
Compression-based fixedparameter algorithms for feedback vertex set and edge bipartization
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- 2011
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Citation Context ...er tractable with respect to the parameter k as defined above. More specifically, there is an algorithm exactly solving Edge Bipartization in O(2 k · m 2 ) time (m denoting the number of graph edges) =-=[8]-=-. For relatively small parameter values k, this opens the way to a viable and efficient alternative to employing approximation algorithms. In this work, we observe that the Balanced Subgraph problem e... |

21 |
Computing Vertex Connectivity: New Bounds from Old Techniques
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Citation Context ...is quite complicated and error-prone to implement (several errors in the original publication have been pointed out [10]). Separators of size k for small k can be found efficiently by flow techniques =-=[12]-=-. However, after some experiments we settled for the subsequently described heuristic instead, which is faster and produces no worse results in our tests. Let N(X) := {u | {u, v} ∈ E ∧ v ∈ X} \ X. For... |

14 | Y.: Algorithmic and complexity results for decompositions of biological networks into monotone subsystems
- DasGupta, Enciso, et al.
- 2007
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Citation Context ...s (see [22] for a bibliography of signed graphs), and has been frequently rediscovered since. In particular, motivated by the mathematical analysis of large-scale biological networks, DasGupta et al. =-=[3]-=- use it to model the concept of “monotone subsystems”, under the name of “sign-consistent graphs”. A graph G = (V, E) with edges labeled by h : E → {0, 1} is balanced if there is a vertex coloring f :... |

14 | Algorithm engineering for optimal graph bipartization
- Hüffner
(Show Context)
Citation Context ...fy the Edge Bipartization algorithm to work for Balanced Subgraph. Further, we employ a heuristic speedup trick similar to the one used for an iterative compression algorithm for Vertex Bipartization =-=[14]-=-. We inferred speedups with this trick up to a factor of about 10 12 . We refer to the full version of this paper for details.sProc. 6th WEA, Vol. 4525 in LNCS, pp. 297–310, Springer, 2007 4 Experimen... |

14 |
Networks with Tunable Degree Distribution and
- Volz
- 2003
(Show Context)
Citation Context ... this network is hard to attack by our data reduction. Random Networks. To further substantiate our experimental results, we generated a test bed of random graphs with the algorithm described by Volz =-=[21]-=-. Thereby, we tried to model the yeast network by choosing the following settings: the cluster coefficient is set to 0.016, the distribution of node-degrees is set to power-law with α = −2.2, and the ... |

11 |
The multi-multiway cut problem
- Avidor, Langberg
- 2004
(Show Context)
Citation Context ...so known as (unweighted) Minimum Uncut, is MaxSNP-hard. Here, the best known approximation algorithm finds in polynomial time a solution of size O(k log k), where k is the size of an optimal solution =-=[1]-=-. It has been conjectured that it is NP-hard to improve the log k-factor to a constant [15]. Hence, with respect to practical applications using a polynomial-time approximation algorithm of the minimi... |

3 | Zelikovsky, Fast and efficient brightfield AAPSM conflict detection and correction
- Chiang, Kahng, et al.
- 2007
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Citation Context ...d Subgraph, called Undirected Labeling Problem by DasGupta et al. [3]. This problem also finds numerous other applications, e. g., in statistical physics and integrated circuit fabrication techniques =-=[2, 22]-=-. ⋆ Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group PIAF (fixed-parameter algorithms, NI 369/4), project PEAL (parameterized complexity and exact algorithms, NI 369/1), a... |

3 |
S.V.: Practical partitioningbased methods for the Steiner problem
- Polzin, Daneshmand
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Citation Context ...ior is examined by exhaustively enumerating possible states of the separator and finding exact solutions to the small component C. A similar method has been suggested by Polzin and Vahdati Daneshmand =-=[19]-=- for the Steiner Tree problem. However, they do not employ gadgets and have no formal characterization of reducible cases. Reduction Scheme. Let S be a separator and let C be a small component obtaine... |

3 |
Bibliography of signed and gain graphs
- Zaslavsky
- 1998
(Show Context)
Citation Context ...reviously only approximations [DasGupta et al., WEA 2006] were known. 1 Introduction The concept of balanced signed graphs was first introduced by Harary [11] for the analysis of social networks (see =-=[22]-=- for a bibliography of signed graphs), and has been frequently rediscovered since. In particular, motivated by the mathematical analysis of large-scale biological networks, DasGupta et al. [3] use it ... |

1 |
Modeling methanogenesis with a genome scale metabolic reconstruction of Methanosarcina barkeri
- Feist, Scholten, et al.
(Show Context)
Citation Context ...receptor [18], we could reduce the number of vertices from 688 to 244 and the number of edges from 2208 to 1159 within three minutes. For the regulatory network of the archaeon Methanosarcina barkeri =-=[5]-=-, we were less successful. The number of vertices was decreased from 628 to 500 and the number of edges from 7302 to 6845 in 25 minutes. This could be a hint that the very dense structure of this netw... |

1 |
Molecular interaction map of a macrophage. Research Report
- Oda, Kimura, et al.
- 2004
(Show Context)
Citation Context ...m to the approximation algorithm of DasGupta et al. [3]. The authors considered the regulatory networks of yeast and human epidermal growth factor (EGFR). We additionally examine a macrophage network =-=[17]-=-. The results of both algorithms are given in Table 1. Apart from giving an optimal solution instead of an approximative one, we can also decrease the running time for the yeast and macrophage network... |