## Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs (1994)

Citations: | 40 - 5 self |

### BibTeX

@MISC{Kaplan94tractabilityof,

author = {Haim Kaplan and Ron Shamir and Robert E. Tarjan},

title = {Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the parameterized complexity of three NP-hard graph completion problems. The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT). The PROPER

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Citation Context ...sk is called a k-proper interval supergraph of G. 2.1 A linear algorithm for fixed k. Denote by c l the l-th Catalan number. i.e. c l = \Gamma 2l l \Delta 1 l+1 . Note that c l ! 4 l . Lemma 2.1 (cf. =-=[29]-=-) The number of minimal triangulations of a cycle with l vertices is c l\Gamma2 . Each such triangulation contains l \Gamma 3 chords. The algorithm will traverse part of a search tree in which each no... |

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Citation Context ...f G. Note that every minimal triangulation of G is represented by at least one leaf. One can find a chordless cycle C in a nonchordal graph with m edges in O(m) time by the MCS algorithm described in =-=[31, 32]-=-. Using the algorithm described in [30] one can generate all minimal triangulations in O(jCj) time for each, observing that there exists a 1-1 correspondence between minimal triangulations of an l-cyc... |

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Citation Context ...ompleteness notions. They also conjectured that each of the containments in the hierarchy is proper. (cf. [1, 9] for definitions and details.) Thus, for example, VERTEX COVER and PATHWIDTH are in FPT =-=[9, 3, 17]-=- but INDEPENDENT SET is W[1]-complete [1], and BANDWIDTH is W[t]-hard for all t [4]. The first parameterized problem which we study here involves chordal graphs. A graph is chordal (or triangulated) i... |

183 |
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Citation Context ...es the number of non-zeros generated in the elimination process (assuming no lucky cancellation of non-zeros) is equivalent to solving the minimum fill-in problem on a corresponding graph. Yannakakis =-=[34]-=- has shown that minimum fill-in is NP-complete. We focus here on the parametrized version of the problem: ffl (FILL-IN(k) Problem:) Given as input a graph, does it have fill-in at most k? Here k is fi... |

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Citation Context ... all minimal triangulations in O(jCj) time for each, observing that there exists a 1-1 correspondence between minimal triangulations of an l-cycle and binary trees with l \Gamma 2 internal nodes (cf. =-=[28]-=-). The nodes of this search tree which are actually traversed correspond to supergraphs of G with no more than k additional edges. If one such node is a leaf then we have found a k-triangulation. Othe... |

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Citation Context ...CHORDAL GRAPH COMPLETION) is to find a fill-in set of minimum cardinality. This problem has received a lot of attention in the past due to its importance in sparse matrix computation (cf. [12]). Rose =-=[26]-=- has shown that for a sparse, symmetric matrix, finding an order of Gaussian elimination steps on diagonal elements which minimizes the number of non-zeros generated in the elimination process (assumi... |

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Citation Context ...ing clones. In the corresponding interval graph G, this is its clique size, denoted !(G). Typical biological maps have width between 5 and 15, even when the total number of clones is in the thousands =-=[18, 23]-=-. Motivated by this observation, we concentrate on two problems: ffl Completion to a proper interval graph with bounded clique size ( Problem A): Given a graph G and a constant k, does there exist a s... |

77 |
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Citation Context ...is a unit interval graph if all intervals assigned have equal length. It is proper interval if it has an assignment in which no interval properly contains another. The last two notions are equivalent =-=[24]-=-. The PROPER INTERVAL GRAPH COMPLETION (PIGC) problem is to find the smallest edge set whose addition to the input graph will form a proper interval graph. Interval completion problems arise in molecu... |

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Citation Context ... t ? 0. E. Problem B is NP-complete and its parametric version is W[t] hard for every t ? 0. These results follow immediately from known results on bandwidth, using Theorem 3.2: Result A follows from =-=[11, 20]-=-, B follows from [14], C from [11] and D from [4]. Result E follows since Problem B is a generalization of Problem A and thus at least as hard. We next describe a polynomial algorithm for Problem B wh... |

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Citation Context ...rized decision problem classes, FPT ` W [1] ` W [2] ` : : : and appropriate reducibility and completeness notions. They also conjectured that each of the containments in the hierarchy is proper. (cf. =-=[1, 9]-=- for definitions and details.) Thus, for example, VERTEX COVER and PATHWIDTH are in FPT [9, 3, 17] but INDEPENDENT SET is W[1]-complete [1], and BANDWIDTH is W[t]-hard for all t [4]. The first paramet... |

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Citation Context ...is proper. (cf. [1, 9] for definitions and details.) Thus, for example, VERTEX COVER and PATHWIDTH are in FPT [9, 3, 17] but INDEPENDENT SET is W[1]-complete [1], and BANDWIDTH is W[t]-hard for all t =-=[4]-=-. The first parameterized problem which we study here involves chordal graphs. A graph is chordal (or triangulated) if every cycle of length four or more contains a chord (an edge between nonadjacent ... |

53 |
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Citation Context ...iguous intervals of the DNA chain (called clones) is given together with experimental information on their pairwise overlaps. The goal is to build a map describing the relative position of the clones =-=[5, 21, 16, 4]. We conce-=-ntrate here on the biologically important case where all clones have equal length. In the presence of "false negative " errors (unidentified overlaps) the problem of building a map with fewe... |

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Citation Context ...alled those parameterized problems which have algorithms of complexity O(f(k)n ff ) fixed parameter tractable (FPT), and defined a hierarchy of parameterized decision problem classes, FPT ` W [1] ` W =-=[2]-=- ` : : : and appropriate reducibility and completeness notions. They also conjectured that each of the containments in the hierarchy is proper. (cf. [1, 9] for definitions and details.) Thus, for exam... |

50 | Computer Solution of Large Sparse Positive De nite Systems, Prentice-Hall, Englewood Cli s - George, Liu - 1981 |

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Citation Context ... t ? 0. E. Problem B is NP-complete and its parametric version is W[t] hard for every t ? 0. These results follow immediately from known results on bandwidth, using Theorem 3.2: Result A follows from =-=[11, 20]-=-, B follows from [14], C from [11] and D from [4]. Result E follows since Problem B is a generalization of Problem A and thus at least as hard. We next describe a polynomial algorithm for Problem B wh... |

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Citation Context ...where all clones have equal length. In the presence of "false negative " errors (unidentified overlaps) the problem of building a map with fewest errors is equivalent to PIGC. This problem i=-=s NP-hard [13]-=-. But what about its complexity for a small fixed number of errors? Let PIGC(k) be the parameterized version of the problem in which one asks for an augmenting set with no more than k edges if one exi... |

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Citation Context ... pathwidth of a graph, originally introduced by Robertson and Seymour [25]. It is well-known that the pathwidth of a graph is one less than the least clique size of any interval supergraph of it (cf. =-=[19]-=-). The following analogous result holds for proper pathwidth: Observation 3.1 ppw(G) = minf!(G 0 )jG ` G 0 and G 0 is proper intervalg \Gamma 1. Thus Problem A is equivalent to deciding whether the pr... |

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- 1984
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Citation Context ...P-complete and its parametric version is W[t] hard for every t ? 0. These results follow immediately from known results on bandwidth, using Theorem 3.2: Result A follows from [11, 20], B follows from =-=[14]-=-, C from [11] and D from [4]. Result E follows since Problem B is a generalization of Problem A and thus at least as hard. We next describe a polynomial algorithm for Problem B when k is fixed. A sand... |

29 | bandwidth and completion problems to proper interval graphs with small cliques - Kaplan, Shamir, et al. - 1996 |

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Citation Context ...es which triangulates GA . Then F also triangulates G. In order to prove this theorem we need the following lemma which can be proved using a characterization of minimal chordal completions proved in =-=[22]-=-. Lemma 2.10 Let ~ G = (V; ~ E) be a minimal chordal completion of a graph G(V; E). Any edge in ~ E \Gamma E is a chord in some chordless cycle of G. Proof. of Theorem 2.9: We need to prove that G = (... |

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Citation Context ...he following observations. Proper interval graphs are exactly the chordal graphs which do not contain any of the three obstructions in Figure 1 as an induced subgraph [33]. Modifying the algorithm of =-=[6]-=- one can find such an obstruction in a chordal graph which is not proper interval in linear time. Using these observations we can show: Theorem 2.3 All k-proper interval supergraphs of a graph can be ... |

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Citation Context ...l(v) for all v 2V. It is easy to show that L is legal and satisfies bwL (S)sk. The algorithm for Problem B uses the equivalence established above. It is an extension of the algorithm for BANDWIDTH in =-=[27, 14]-=- to our problem, recast as deciding if bw(S)sk \Gamma 1. We assume w.l.o.g. that in the input sandwich instancesS each vertex is incident on no more than 2k E r -edges, and that G r = (V; E r ) is con... |

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Citation Context ...n k may be crucial for applications where small, fixed parameter values are important, as in the problems we study here. Downey and Fellows initiated a systematic complexity analysis of such problems =-=[7, 1, 8]-=-. They called those parameterized problems which have algorithms of complexity O(f(k)n ff ) fixed parameter tractable (FPT), and defined a hierarchy of parameterized decision problem classes, FPT ` W ... |

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Citation Context ...iguous intervals of the DNA chain (called clones) is given together with experimental information on their pairwise overlaps. The goal is to build a map describing the relative position of the clones =-=[5, 21, 16, 4]. We conce-=-ntrate here on the biologically important case where all clones have equal length. In the presence of "false negative " errors (unidentified overlaps) the problem of building a map with fewe... |

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Eigenschaften der Nerven homologisch-einfacher Familien im R n
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Citation Context ...al supergraphs of a graph using the following observations. Proper interval graphs are exactly the chordal graphs which do not contain any of the three obstructions in Figure 1 as an induced subgraph =-=[33]-=-. Modifying the algorithm of [6] one can find such an obstruction in a chordal graph which is not proper interval in linear time. Using these observations we can show: Theorem 2.3 All k-proper interva... |

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Citation Context ...by a factor of n in the space complexity of the algorithm. Obtaining from the layout a proper interval assignment can subsequently done in O(n) time. A biologically-motivated restriction of Problem B =-=[10, 4]-=- is defined by the graph G and a k-coloring c of it, and the requirement that the set of added edges must not violate the coloring. Hence E f is implicitly defined by f(u; v)jc(u) = c(v)g. We call thi... |

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13 | Tarjan, Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs - Kaplan, Shamir, et al. - 1906 |