## Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques (1996)

Venue: | SIAM Journal on Computing |

Citations: | 30 - 6 self |

### BibTeX

@ARTICLE{Kaplan96pathwidth,bandwidth,

author = {Haim Kaplan and Ron Shamir},

title = {Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques},

journal = {SIAM Journal on Computing},

year = {1996},

volume = {25},

pages = {540--561}

}

### OpenURL

### Abstract

We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper k-coloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NP-hard, and W [t]-hard for all t. We also show that the second problem is W [1]-hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.

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Citation Context ...es In this section we give definitions and background, and state without proof some well-known facts about notions which we will need in the sequel. For other graph theoretical definitions see, e.g., =-=[19]-=-. Basic definitions: All graphs are assumed to be undirected, simple and finite. A graph G = (V; E) is a supergraph of the graph G 0 = (V 0 ; E 0 ) if V = V 0 and E ' E 0 . For a subset W ` V , the su... |

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Citation Context ...nsists of those parameterized decision problems which parametrically reduce to \Pi F , where F is a family of mixed boolean decision circuits of bounded depth and weft at most t. Thus FPT ` W [1] ` W =-=[2]-=- ` : : : is the W -hierarchy. All containments are conjectured to be proper [11]. A problem \Pi is W [s]-hard if every problem in W [s] has a parameterized reduction to \Pi. If \Pi is both in W [s] an... |

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Citation Context ...G) j L is a layout of Gg. The BANDWIDTH problem is to decide for a given graph G and integer k, if bw(G)sk. This problem has been studied intensely because of its application to sparse matrix algebra =-=[9]-=-. It is known to be NP-complete even for binary trees [16] and for caterpillars with hair length at most three [34]. On the other hand, it is solvable in O(n k ) for arbitrary k [23], and in linear ti... |

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Citation Context ... if the pathwidth of G is at most k \Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. [33]). It is NP-hard [28, 2, 35, 24] but linearly solvable for fixed k =-=[3, 30]-=-. Problem C 0 (and hence also B 0 ) was shown to be NP-hard when k = jV j independently in [21] and [13]. Fellows et al. [13] have also shown that the Problem C 0 is hard for W [1], and that it is not... |

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Citation Context ... data is that the "width" of the map is consistently very small: The largest number of mutually overlapping clones is typically between 5 and 15, compared to a total number of clones in the =-=thousands [31, 37]-=-. Our study here focuses on the tractability of the models above in this special situation. More formally, do the problems remain NP-hard when the clique size of the proper interval graph is assumed t... |

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Citation Context ...s the minimum value of pwX (G) over all path decompositions, i.e., pw(G) = minfpwX (G) j X is a path decomposition of Gg. The notion of pathwidth was originally introduced by Robertson and Seymour in =-=[39]-=-. Lemma 2.3 (e.g., [6]) For any path decomposition X = (X 1 ; : : : ; X l ) of a graph G, every clique in G must be contained in some X i . The PATHWIDTH problem is to decide for a given graph G and a... |

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Citation Context ... (or ambiguity) of the problem data, where the edges of E 2 \Gamma E 1 are not known for sure to be included or excluded from the graph. Sandwich problems were introduced in [22] and studied later in =-=[21, 20, 5, 13]-=-. Their applications include matrix algebra, evolutionary tree construction and temporal reasoning. Physical Mapping of DNA 6 motivated our study here of the following problem: PROPER INTERVAL SANDWIC... |

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Citation Context ...hm with time complexity bounded by f(k)n ff for any constant ff. Problem B may be viewed as a sandwich problem, since G 0 must be "sandwiched" between G and G 2 . Sandwich problems were intr=-=oduced in [22]-=-. Problem B was shown to be NP-hard for the case where k = jV j (i.e., without any restriction on the clique size) in [21]. The NPhardness results here are stronger since they apply to problem C, a re... |

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Citation Context ...ide for a given graph G and integer k, if bw(G)sk. This problem has been studied intensely because of its application to sparse matrix algebra [9]. It is known to be NP-complete even for binary trees =-=[16]-=- and for caterpillars with hair length at most three [34]. On the other hand, it is solvable in O(n k ) for arbitrary k [23], and in linear time for k = 2 [16]. Sandwich Problems: Given two graphs G 1... |

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Citation Context ...pwX (G) over all path decompositions, i.e., pw(G) = minfpwX (G) j X is a path decomposition of Gg. The notion of pathwidth was originally introduced by Robertson and Seymour in [39]. Lemma 2.3 (e.g., =-=[6]-=-) For any path decomposition X = (X 1 ; : : : ; X l ) of a graph G, every clique in G must be contained in some X i . The PATHWIDTH problem is to decide for a given graph G and a given integer k if pw... |

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Citation Context ...lynomial when the parameter k is fixed. We prove that problem A is equivalent to the BANDWIDTH problem. In particular, this equivalence together with a recent result of Bodlaender, Fellows and Hallet =-=[4]-=- imply that Problems A and B are hard for the parameterized complexity class W [t], for all t. We prove here that Problem C is W [1]-hard. This implies that none of the problems is likely to have an a... |

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Citation Context ...mize the number of edges whose addition to the graph will form an interval graph. If one assumes that all errors are erroneous overlaps (false positives), the interval graph deletion problem comes up =-=[18]-=-. If some pairs of clones are definite overlaps, some are definite non-overlaps and the rest are unknown then one gets the interval sandwich problems[22]. All three problems are NP-hard, for interval ... |

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Citation Context ...nformation on their relative position along the DNA chain is lost. The goal of physical mapping of DNA is to reconstruct that order, based on experimental data on the overlaps between pairs of clones =-=[8, 36, 26]-=-. Hence, the graph with vertices corresponding to clones and edges corresponding to overlapping pairs of clones should be an interval graph if clone lengths vary, or a proper interval graph if all clo... |

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Citation Context ... (or ambiguity) of the problem data, where the edges of E 2 \Gamma E 1 are not known for sure to be included or excluded from the graph. Sandwich problems were introduced in [22] and studied later in =-=[21, 20, 5, 13]-=-. Their applications include matrix algebra, evolutionary tree construction and temporal reasoning. Physical Mapping of DNA 6 motivated our study here of the following problem: PROPER INTERVAL SANDWIC... |

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Citation Context ...uence is the equivalence of this parameter to the bandwidth of the graph. Portions of this paper were presented at the 34th Annual IEEE Symp. on the Foundations of Computer Science, Santa Fe, NM 1994 =-=[25]-=- y Research supported in part by a grant from the Ministry of Science and Technology, Israel. email: shamir@math.tau.ac.il 1 Introduction This paper studies the following graph theoretic questions: ff... |

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Citation Context ... interval instead of unit interval. Problem A 0 is equivalent to asking if the pathwidth of G is at most k \Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. =-=[33]-=-). It is NP-hard [28, 2, 35, 24] but linearly solvable for fixed k [3, 30]. Problem C 0 (and hence also B 0 ) was shown to be NP-hard when k = jV j independently in [21] and [13]. Fellows et al. [13] ... |

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Citation Context ...must be "sandwiched" between G and G 2 . Sandwich problems were introduced in [22]. Problem B was shown to be NP-hard for the case where k = jV j (i.e., without any restriction on the clique=-= size) in [21]-=-. The NPhardness results here are stronger since they apply to problem C, a restriction of problem B in which the forbidden edges are those between like-colored endpoints. Let A 0 ; B 0 and C 0 be the... |

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Citation Context ...vertex degrees at most three [35]. On the other hand, it is polynomial when k is fixed. The results of Robertson and Seymour [39] imply non-constructively that an O(n 2 ) algorithm exists for fixed k =-=[14, 15]-=-. The recent results of Bodlaender [3] yield a linear algorithm for PATHWIDTH for every fixed k [30, chapter 11]. The notions of pathwidth and interval graphs are related by the following well-known o... |

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Citation Context ... to overlapping pairs of clones should be an interval graph if clone lengths vary, or a proper interval graph if all clones have the same length. Both types of graphs can be recognized in linear time =-=[7, 32, 10]-=-. In practice, information on overlap is never perfect. There are several ways to model this imperfection: If one assumes that all the errors in the data are false non-overlaps (false negatives) and w... |

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Citation Context ... k\Gamma1 ) steps. Hence, Problems B and C are polynomial when k is fixed. Our algorithm is based on the equivalence established in the previous sections with the bandwidth problem. For fixed k, Saxe =-=[40]-=- gave an O(n k+1 ) algorithm which determines if the bandwidth of an n-vertex graph is at most k. Gurari and Sudborough [23] reduced its complexity to O(n k ). We generalize that algorithm to the prob... |

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Citation Context ...unit interval. Problem A 0 is equivalent to asking if the pathwidth of G is at most k \Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. [33]). It is NP-hard =-=[28, 2, 35, 24]-=- but linearly solvable for fixed k [3, 30]. Problem C 0 (and hence also B 0 ) was shown to be NP-hard when k = jV j independently in [21] and [13]. Fellows et al. [13] have also shown that the Problem... |

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Citation Context ...nformation on their relative position along the DNA chain is lost. The goal of physical mapping of DNA is to reconstruct that order, based on experimental data on the overlaps between pairs of clones =-=[8, 36, 26]-=-. Hence, the graph with vertices corresponding to clones and edges corresponding to overlapping pairs of clones should be an interval graph if clone lengths vary, or a proper interval graph if all clo... |

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Citation Context ...ection: If one assumes that all the errors in the data are false non-overlaps (false negatives) and wishes to minimize their number, one gets the interval graph completion problem [17, problem GT35], =-=[27]-=-: Minimize the number of edges whose addition to the graph will form an interval graph. If one assumes that all errors are erroneous overlaps (false positives), the interval graph deletion problem com... |

18 |
DNA physical mapping: Three ways difficult
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Citation Context ...n numerous areas (cf. [33]). It is NP-hard [28, 2, 35, 24] but linearly solvable for fixed k [3, 30]. Problem C 0 (and hence also B 0 ) was shown to be NP-hard when k = jV j independently in [21] and =-=[13]-=-. Fellows et al. [13] have also shown that the Problem C 0 is hard for W [1], and that it is not finite state for bounded pathwidth or treewidth, and hence not polynomially solvable by conventional al... |

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Citation Context ...n E 3 = ;, every layout of S is legal and bw(S) = bw(G 1 ). Parameterized Complexity: Recently, Downey and Fellows initiated a systematic analysis of the complexity of parameterized decision problems =-=[11, 1, 12]-=-. An instance of a parameterized decision problem \Pi is a pair (x; k) where k is the parameter and x is the input other than the parameter, with jxj = n. The interest is usually in parameterized prob... |

17 |
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Citation Context ...unit interval. Problem A 0 is equivalent to asking if the pathwidth of G is at most k \Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. [33]). It is NP-hard =-=[28, 2, 35, 24]-=- but linearly solvable for fixed k [3, 30]. Problem C 0 (and hence also B 0 ) was shown to be NP-hard when k = jV j independently in [21] and [13]. Fellows et al. [13] have also shown that the Problem... |

17 |
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Citation Context ...unit interval. Problem A 0 is equivalent to asking if the pathwidth of G is at most k \Gamma 1, and it arises in various guises (and under different names) in numerous areas (cf. [33]). It is NP-hard =-=[28, 2, 35, 24]-=- but linearly solvable for fixed k [3, 30]. Problem C 0 (and hence also B 0 ) was shown to be NP-hard when k = jV j independently in [21] and [13]. Fellows et al. [13] have also shown that the Problem... |

14 |
An incremental linear time algorithm for recognizing interval graphs
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Citation Context ... to overlapping pairs of clones should be an interval graph if clone lengths vary, or a proper interval graph if all clones have the same length. Both types of graphs can be recognized in linear time =-=[7, 32, 10]-=-. In practice, information on overlap is never perfect. There are several ways to model this imperfection: If one assumes that all the errors in the data are false non-overlaps (false negatives) and w... |

8 |
Fixed-Parameter Intractability II
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Citation Context ...r hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NP-hard, and W [t]-hard for all t. We also show that the second problem is W =-=[1]-=--hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ff ) algorithm, where ff is a constant independent of k. A central tool in our study is a new graph theoretic par... |

8 |
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Citation Context ... data is that the "width" of the map is consistently very small: The largest number of mutually overlapping clones is typically between 5 and 15, compared to a total number of clones in the =-=thousands [31, 37]-=-. Our study here focuses on the tractability of the models above in this special situation. More formally, do the problems remain NP-hard when the clique size of the proper interval graph is assumed t... |

5 |
Current approaches to long-range physical mapping of the humon genome
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(Show Context)
Citation Context ...nformation on their relative position along the DNA chain is lost. The goal of physical mapping of DNA is to reconstruct that order, based on experimental data on the overlaps between pairs of clones =-=[8, 36, 26]-=-. Hence, the graph with vertices corresponding to clones and edges corresponding to overlapping pairs of clones should be an interval graph if clone lengths vary, or a proper interval graph if all clo... |

3 |
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(Show Context)
Citation Context ...vertex degrees at most three [35]. On the other hand, it is polynomial when k is fixed. The results of Robertson and Seymour [39] imply non-constructively that an O(n 2 ) algorithm exists for fixed k =-=[14, 15]-=-. The recent results of Bodlaender [3] yield a linear algorithm for PATHWIDTH for every fixed k [30, chapter 11]. The notions of pathwidth and interval graphs are related by the following well-known o... |