## A polyhedral approach to sequence alignment problems (2000)

Venue: | DISCRETE APPL. MATH |

Citations: | 21 - 1 self |

### BibTeX

@ARTICLE{Kececioglu00apolyhedral,

author = {John D. Kececioglu and Hans-peter Lenhof and Kurt Mehlhorn and Petra Mutzel and Knut Reinert and Martin Vingron},

title = {A polyhedral approach to sequence alignment problems},

journal = {DISCRETE APPL. MATH},

year = {2000},

pages = {143--186}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study two new problems in sequence alignment both from a practical and a theoretical view, using tools from combinatorial optimization to develop branch-and-cut algorithms. The Generalized Maximum Trace formulation captures several forms of multiple sequence alignment problems in a common framework, among them the original formulation of Maximum Trace. The RNA Sequence Alignment Problem captures the comparison of RNA molecules on the basis of their primary sequence and their secondary structure. Both problems have a characterization in terms of graphs which we reformulate in terms of integer linear programming. We then study the polytopes (or convex hulls of all feasible solutions) associated with the integer linear program for both problems. For each polytope we derive several classes of facet-defining inequalities and show that for some of these classes the corresponding separation problem can be solved in polynomial time. This leads to a polynomial time algorithm for pairwise sequence alignment that is not based on dynamic programming. Moreover, for multiple sequences the branch-and-cut algorithms for both sequence alignment problems are able to solve to optimality instances that are beyond the range of present dynamic programming approaches.

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Citation Context ...e sequences in order to exhibit their commonalities. It is interesting that while the diversity of alignment problems and their associated algorithms has grown tremendously since Needleman and Wunsch =-=[28]-=- first published their paper on two-sequence alignment in 1970, most alignment problems that have been studied have been solved by dynamic programming. This technique, while quite powerful, has the dr... |

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Citation Context ...ubproblems. 5 Computational results In this section we report on the results generated by our program. The implementation is coded in C++ using the library of efficient data types and algorithms LEDA =-=[25]-=- and the branch-and-cut framework ABACUS [17]. 5.1 The GMT problem We tested three different ways to generate the extended alignment graph. • As an example of a scoring scheme based on the comparison ... |

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Citation Context ...0 > 0. The above theorem for full-dimensional polytopes restricts the number of possible facet-defining inequalities. The next theorem will prove useful in the two-sequence case of the GMT. Theorem 5 =-=[30]-=- Suppose F ⊆ A is a maximal clique in the k-regular independence system (A, I). Then ∑ e∈F xe ≤ k − 1 is a facet of PI. For F ⊆ A we call I ′ = (F, I ′ ) where I ′ = {C ∈ I| C ⊆ F }, the subsystem gen... |

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Citation Context ...hm is a set of reasonable alignment edges. In principle, we use for this purpose alignment edges realized by some suboptimal alignment, i.e. an alignment with a score close to optimal. In contrast to =-=[21]-=- we do not take all the edges realized by any suboptimal alignments scoring better than a fixed threshold s below the optimal. Rather we employ a windowing technique to make the alignment graph denser... |

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Citation Context ...d newly-discovered classes. 2Graphs, traces and multiple alignment To describe the GMT and RSA problem we first review a formulation of multiple alignment in terms of graphs introduced by Kececioglu =-=[19]-=- and show how to extend this formulation to model the two new problems. C C G e f g h G C G (a) U - - C f G C C - (b) G h G U Fig. 1. (a) An alignment graph of two sequences CGC and GCGU. Edges e, f a... |

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Citation Context ...ures more general scoring schemes based on the similarity of pairs of whole segments of the sequences pairs (see for instance Altschul and Erickson [1], Morgenstern et al. [26], and Wilbur and Lipman =-=[41]-=-). To illustrate how this can be done, Figure 3 shows a partition into sets of edges that form consecutive runs of matches. Here the edges of a run from a block. 4Note that the two blocks d5 and d1 e... |

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Citation Context ... the Generalized Maximum Trace (GMT) and RNA Sequence Alignment (RSA) problems yields an algorithm for each problem that is not based on dynamic programming but is known as a branch-and-cut algorithm =-=[16]-=-. Branch-and-cut algorithms combine linear programming with the branch-and-bound paradigm, and are currently the most successful algorithms for solving hard combinatorial problems such as the famous T... |

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Citation Context ...re able to solve problem instances to optimality, the size of which is not tractable for dynamic programming based approaches. Sophisticated implementations of such approaches such as MSA [12] or GSA =-=[22]-=- cannot possibly solve non-trivial problem instances of 18 sequences. This is due to the exponential space consumption of dynamic programming. Although both programs can compute an alignment of guaran... |

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Citation Context ...m-of-pairs multiple alignment problem. GMT also captures more general scoring schemes based on the similarity of pairs of whole segments of the sequences pairs (see for instance Altschul and Erickson =-=[1]-=-, Morgenstern et al. [26], and Wilbur and Lipman [41]). To illustrate how this can be done, Figure 3 shows a partition into sets of edges that form consecutive runs of matches. Here the edges of a run... |

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Citation Context ...ble to real-life problems. Gorodkin et al. [8] iterate Sankoff’s dynamic programming algorithm to find motifs among many RNA sequences.” Instead of using dynamic programming the algorithm of Waterman =-=[40]-=- searches for common motifs among several sequences. Eddy and Durbin [7] de5scribe probabilistic models for measuring the secondary structure and primary sequence consensus of RNA sequence families. ... |

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Citation Context ...ection we report on the results generated by our program. The implementation is coded in C++ using the library of efficient data types and algorithms LEDA [25] and the branch-and-cut framework ABACUS =-=[17]-=-. 5.1 The GMT problem We tested three different ways to generate the extended alignment graph. • As an example of a scoring scheme based on the comparison of two residues (MT) we adapted the PRIMAL [2... |

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Citation Context ...s combine linear programming with the branch-and-bound paradigm, and are currently the most successful algorithms for solving hard combinatorial problems such as the famous Traveling Salesman Problem =-=[15,2]-=-. We view as one of the contributions of our work the introduction of the polyhedral approach to the area of sequence alignment, and our experience with these relatively new techniques has helped us t... |

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Citation Context ...nment problem. GMT also captures more general scoring schemes based on the similarity of pairs of whole segments of the sequences pairs (see for instance Altschul and Erickson [1], Morgenstern et al. =-=[26]-=-, and Wilbur and Lipman [41]). To illustrate how this can be done, Figure 3 shows a partition into sets of edges that form consecutive runs of matches. Here the edges of a run from a block. 4Note tha... |

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Citation Context ...paration problem can be solved in polynomial time. This leads to another polynomial time algorithm for pairwise alignment that is not based on dynamic programming techniques (see Pevzner and Waterman =-=[33]-=- for a thorough presentation of primal-dual approach for a number of sequence alignment problems that is also not based on dynamic programming). It turns out that our branch-and-cut algorithms can sol... |

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Citation Context ...o on to Section 4 in which the algorithms are described. As a final note the corresponding author would like to point out that a more detailed discussion of the problems in this paper can be found in =-=[34]-=-. 2 A graph-theoretic characterization of traces In this section we give a graph-theoretic characterization of traces in a form that is helpful for expressing the GMT and RSA problem as integer linear... |

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Citation Context ...neralized to multiple sequence alignment with the notion of a trace of an alignment graph. The relationship between multiple alignment and multipartite graphs was also examined by Vingron and Pevzner =-=[39]-=- in the context of filtering pairwise dot-plots of a set of sequences. 3The Generalized Maximum Trace Problem In the Maximum Trace Problem (MT), introduced originally to model the final multiple alig... |

2 |
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Citation Context ...7]. 5.1 The GMT problem We tested three different ways to generate the extended alignment graph. • As an example of a scoring scheme based on the comparison of two residues (MT) we adapted the PRIMAL =-=[20]-=- package by John Kececioglu. The value of the approximate solution of this program is used as a lower bound in our branch-and-cut algorithm. • As an example for a scoring scheme based on the compariso... |

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Citation Context ... k-regular system (A, I) if |F | ≥ k and all ( |F | k k-subsets of F are circuits of (A, I). Let PI be the polyhedron associated with (A, I). Then the following well known theorems hold. 13Theorem 3 =-=[11]-=- Let (A, I) be an independence system and let F = A − ⋃ I. Then the dimension of PI is |A| − |F |. This theorem yields a method to determine whether a polytope is fulldimensional. Theorem 4 [13] If PI... |