## NC Algorithms for Comparability Graphs, Interval Graphs, and Unique Perfect Matchings (1985)

Venue: | In Proceedings of FST&TCS Conference, LNCS Volume 206 |

Citations: | 12 - 0 self |

### BibTeX

@INPROCEEDINGS{Kozen85ncalgorithms,

author = {Dexter C. Kozen and Umesh V. Vazirani and Vijay V. Vazirani},

title = {NC Algorithms for Comparability Graphs, Interval Graphs, and Unique Perfect Matchings},

booktitle = {In Proceedings of FST&TCS Conference, LNCS Volume 206},

year = {1985},

pages = {496--503},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Laszlo Lovasz recently posed the following problem: "Is there an NC algorithm for testing if a given graph has a unique perfect matching ?" We present such an algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for finding a maximum matching in an incomparability graph. 1 Introduction Karp, Upfal and Wigderson [9] have recently shown that the maximum matching problem is in Random NC 3 (RNC 3 ). This result has since been improved to RNC 2 by Mulmuley, Vazirani, and Vazirani [16]. It remains open whether there is a deterministic NC algorithm for this problem. A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz [13] (see [1]). Rabin and Vazirani [18] give an NC algorithm for obtaining perfect matchings in...

### Citations

381 | A simple parallel algorithm for the maximal independent set problem
- Luby
- 1986
(Show Context)
Citation Context ... the interval ff(v). This will yield the set of intervals I. To accomplish step 2, we can use the parallel maximal clique algorithm of Karp and Wigderson [10], or the more efficient algorithm of Luby =-=[14]-=-. We thus obtain: Theorem 32 There is an NC algorithm which checks if a given graph G = (V; E) is an interval graph, and if so obtains an interval representation for it. 5 Parallel Matching Algorithms... |

95 |
Transitiv orientierbare Graphen
- Gallai
- 1967
(Show Context)
Citation Context ...rem that allows a graph to be decomposed according to its comparability structure, allowing separate parts of the graph to be oriented independently. This decomposition was first discovered by Gallai =-=[4]-=-; see Kelly [11] for an excellent account in English. Considerations of efficiency require us to be more careful in the development of the Gallai decomposition. Our development, which we give below in... |

91 |
1964 “A characterization of comparability graphs and of interval graphs
- Gilmore, Hoffman
(Show Context)
Citation Context ...ansitive orientation of a given comparability graph, and an interval representation for a given interval graph. The first sequential algorithms for these problems were obtained by Gilmore and Hoffman =-=[6]-=- and Ghouila-Houri [5]. More efficient algorithms were obtained by Even, Pnueli, and Lempel [2, 17] and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor schedulin... |

83 | A fast parallel algorithm for the maximal independent set problem
- Karp, Wigderson
- 1985
(Show Context)
Citation Context .... Do in parallel for each vertex v 2 G: obtain the interval ff(v). This will yield the set of intervals I. To accomplish step 2, we can use the parallel maximal clique algorithm of Karp and Wigderson =-=[10]-=-, or the more efficient algorithm of Luby [14]. We thus obtain: Theorem 32 There is an NC algorithm which checks if a given graph G = (V; E) is an interval graph, and if so obtains an interval represe... |

61 |
On determinants, matchings, and random algorithms
- Lovasz
- 1979
(Show Context)
Citation Context ...tic NC algorithm for this problem. A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz =-=[13] (see-=- [1]). Rabin and Vazirani [18] give an NC algorithm for obtaining perfect matchings in graphs having a unique perfect matching. Laszlo Lovasz recently posed the following problem: "Is there an NC... |

57 |
Algorithmic aspects of comparability graphs and interval graphs
- Möhring
- 1984
(Show Context)
Citation Context ...English. Considerations of efficiency require us to be more careful in the development of the Gallai decomposition. Our development, which we give below in full, departs from the standard development =-=[11, 15] in s-=-everal technical respects, perhaps the most important of which is the inclusion of edges of the complementary graph in the definition of the relation ", the symmetry of G and G c , and the use of... |

41 | Fast parallel matrix and GCD computations
- GATHEN, J, et al.
- 1982
(Show Context)
Citation Context ...orithm for this problem. A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz [13] (see =-=[1]). Ra-=-bin and Vazirani [18] give an NC algorithm for obtaining perfect matchings in graphs having a unique perfect matching. Laszlo Lovasz recently posed the following problem: "Is there an NC algorith... |

37 |
Transitive orientation of graphs and identification of permutation graphs
- Pnueli, Even, et al.
- 1971
(Show Context)
Citation Context ... interval graph. The first sequential algorithms for these problems were obtained by Gilmore and Hoffman [6] and Ghouila-Houri [5]. More efficient algorithms were obtained by Even, Pnueli, and Lempel =-=[2, 17]-=- and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor scheduling algorithm of Helmbold and Mayr [8] give an NC algorithm for maximum matching in comparability gra... |

29 |
Optimal sequencing of two equivalent processors
- Fujii, Kasami, et al.
- 1969
(Show Context)
Citation Context ...nit on the two processors are said to be paired . The connection between this problem and the maximum matching problem is established by the following theorem: Theorem 33 (Fujii, Kasami, and Ninomiya =-=[3]-=-) Let G = (V; E) be a directed, acyclic graph, and let G c be the complement of its transitive closure. Then the paired jobs in an optimal schedule for G form a maximum matching in G c . This theorem ... |

28 |
Permutation graphs and transitive graphs
- Even, Pnueli, et al.
- 1972
(Show Context)
Citation Context ... interval graph. The first sequential algorithms for these problems were obtained by Gilmore and Hoffman [6] and Ghouila-Houri [5]. More efficient algorithms were obtained by Even, Pnueli, and Lempel =-=[2, 17]-=- and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor scheduling algorithm of Helmbold and Mayr [8] give an NC algorithm for maximum matching in comparability gra... |

24 |
Caractérisation des graphes non orientès dont on peut orienter les arrêtes de maniere à obtenir le graphe d’un relation d’ordre, C.R.Acad. Sci.Paris 254
- Ghouilà-Houri
- 1962
(Show Context)
Citation Context ...f a given comparability graph, and an interval representation for a given interval graph. The first sequential algorithms for these problems were obtained by Gilmore and Hoffman [6] and Ghouila-Houri =-=[5]-=-. More efficient algorithms were obtained by Even, Pnueli, and Lempel [2, 17] and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor scheduling algorithm of Helmbol... |

23 |
Maximum matchings in general graphs through randomization
- Rabin, Vazirani
- 1989
(Show Context)
Citation Context ...A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz [13] (see [1]). Rabin and Vazirani =-=[18] give-=- an NC algorithm for obtaining perfect matchings in graphs having a unique perfect matching. Laszlo Lovasz recently posed the following problem: "Is there an NC algorithm for A preliminary versio... |

17 |
Comparability graphs
- Kelly
- 1985
(Show Context)
Citation Context ... a graph to be decomposed according to its comparability structure, allowing separate parts of the graph to be oriented independently. This decomposition was first discovered by Gallai [4]; see Kelly =-=[11]-=- for an excellent account in English. Considerations of efficiency require us to be more careful in the development of the Gallai decomposition. Our development, which we give below in full, departs f... |

11 |
Constructing a maximum matching is in random NC
- Karp, Upfal, et al.
- 1986
(Show Context)
Citation Context ...ph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for finding a maximum matching in an incomparability graph. 1 Introduction Karp, Upfal and Wigderson =-=[9]-=- have recently shown that the maximum matching problem is in Random NC 3 (RNC 3 ). This result has since been improved to RNC 2 by Mulmuley, Vazirani, and Vazirani [16]. It remains open whether there ... |

7 |
Transitive orientation of graphs and identi cation of permutation graphs
- Pnueli, Even, et al.
- 1971
(Show Context)
Citation Context ... given interval graph. The rst sequential algorithms for these problems were obtained by Gilmore and Ho man [6] and Ghouila-Houri [5]. More e cient algorithms were obtained byEven, Pnueli, and Lempel =-=[2, 17]-=- and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor scheduling algorithm of Helmbold and Mayr [8] give anNC algorithm for maximum matching in comparability grap... |

5 | Comparability graphs and a new matroid - Golumbic - 1977 |

5 |
Two processor scheduling is
- Hembold, Mayr
- 1986
(Show Context)
Citation Context ...cient algorithms were obtained by Even, Pnueli, and Lempel [2, 17] and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor scheduling algorithm of Helmbold and Mayr =-=[8]-=- give an NC algorithm for maximum matching in comparability graphs, a class of graphs containing all interval graphs. This suggests that the maximum matching problem may be in NC . 2 Testing for Uniqu... |

2 |
The two-processor scheduling problem is in random NC
- Vazirani, Vazirani
- 1985
(Show Context)
Citation Context ... in G c . This theorem enabled [3] to obtain a fast sequential algorithm for twoprocessor scheduling, using a maximum matching algorithm as a subroutine. We will do the reverse. Vazirani and Vazirani =-=[19]-=- gave a Random NC algorithm for the twoprocessor scheduling problem, using the RNC matching algorithm of Karp, Upfal, and Wigderson [9] as a subroutine. This gave a weaker version of Theorem 34 below ... |

1 | Vijay Vazirani. NC algorithms for comparability graphs, interval graphs, and unique perfect matching - Kozen, Vazirani - 1985 |

1 |
A characterization of comparability graphs andofinterval graphs
- Gilmore, man
- 1964
(Show Context)
Citation Context ... transitive orientation of a given comparability graph, and an interval representation for a given interval graph. The rst sequential algorithms for these problems were obtained by Gilmore and Ho man =-=[6]-=- and Ghouila-Houri [5]. More e cient algorithms were obtained byEven, Pnueli, and Lempel [2, 17] and Golumbic [7]. The NC comparability graph algorithm, together with the NC two-processor scheduling a... |

1 |
Finding a maximum matching is in random NC .InProc. 26th Symp. Theory of Computing
- Karp, Upfal, et al.
- 1985
(Show Context)
Citation Context ...raph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for nding a maximum matching in an incomparability graph. 1 Introduction Karp, Upfal and Wigderson =-=[9]-=- have recently shown that the maximum matching problem is in Random NC 3 (RNC 3 ). This result has since been improved to RNC 2 by Mulmuley, Vazirani, and Vazirani [16]. It remains open whether there ... |