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A Polyhedral Approach to RNA Sequence Structure Alignment
 Journal of Computational Biology
, 1998
"... Ribonucleic acid (RNA) is a polymer composed of four bases denoted A, C, G, and U. It generally is a singlestranded molecule where the bases form hydrogen bonds within the same molecule leading to structure formation. In comparing different homologous RNA molecules it is important to consider bo ..."
Abstract

Cited by 36 (4 self)
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Ribonucleic acid (RNA) is a polymer composed of four bases denoted A, C, G, and U. It generally is a singlestranded molecule where the bases form hydrogen bonds within the same molecule leading to structure formation. In comparing different homologous RNA molecules it is important to consider both the base sequence and the structure of the molecules. Traditional alignment algorithms can only account for the sequence of bases, but not for the base pairings. Considering the structure leads to significant computational problems because of the dependencies introduced by the base pairings. In this paper we address the problem of optimally aligning a given RNA sequence of unknown structure to one of known sequence and structure. We phrase the problem as an integer linear program and then solve it using methods from polyhedral combinatorics. In our computational experiments we could solve large problem instances  23S ribosomal RNA with more than 1400 bases  a size intractable f...
A polyhedral approach to sequence alignment problems
 DISCRETE APPL. MATH
, 2000
"... We study two new problems in sequence alignment both from a practical and a theoretical view, using tools from combinatorial optimization to develop branchandcut algorithms. The Generalized Maximum Trace formulation captures several forms of multiple sequence alignment problems in a common framewo ..."
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Cited by 20 (1 self)
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We study two new problems in sequence alignment both from a practical and a theoretical view, using tools from combinatorial optimization to develop branchandcut 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 facetdefining 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 branchandcut algorithms for both sequence alignment problems are able to solve to optimality instances that are beyond the range of present dynamic programming approaches.
An Exact Solution for the SegmenttoSegment Multiple Sequence Alignment Problem
"... In molecular biology sequence alignment is a crucial tool in studying structure and function of molecules as well as evolution of species. In the segmenttosegment variation of the multiple alignment problem the input can be seen as a set of runs of nongapped matches (diagonals) between pairs o ..."
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Cited by 17 (8 self)
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In molecular biology sequence alignment is a crucial tool in studying structure and function of molecules as well as evolution of species. In the segmenttosegment variation of the multiple alignment problem the input can be seen as a set of runs of nongapped matches (diagonals) between pairs of sequences. Given a weight function that assigns a weight score to every possible diagonal, the goal is to choose a consistent set of diagonals of maximum weight. We show that the segmenttosegment multiple alignment problem is equivalent to a novel formulation of the Maximum Weight Trace (MWT) problem. Solving the generalized MWT (GMWT) problem to optimality therefore improves upon the previous greedy strategies that are used for solving the segmenttosegment multiple sequence alignment problem. We show that the GMWT can be stated in terms of an integer linear program and then solve the integer linear program using methods from polyhedral combinatorics. This leads to a branchand...
AGDLibrary: A Library of Algorithms for Graph Drawing
, 1997
"... A graph drawing algorithm produces a layout of a graph in two or threedimensional space that should be readable and easy to understand. Since the aesthetic criteria differ from one application area to another, it is unlikely that a definition of the "optimal drawing" of a graph in a strict math ..."
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Cited by 12 (4 self)
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A graph drawing algorithm produces a layout of a graph in two or threedimensional space that should be readable and easy to understand. Since the aesthetic criteria differ from one application area to another, it is unlikely that a definition of the "optimal drawing" of a graph in a strict mathematical sense exists. A large number of graph drawing algorithms taking different aesthetic criteria into account have already been proposed. In this paper we describe the design and implementation of the AGDLibrary, a library of Algorithms for Graph Drawing. The library offers a broad range of existing algorithms for twodimensional graph drawing and tools for implementing new algorithms. The library is written in C++ using the LEDA platform for combinatorial and geometric computing ([16, 17]). The algorithms are implemented independently of the underlying visualization or graphics system by using a generic layout interface. Most graph drawing algorithms place a set of restriction...
TwoLayer Planarization in Graph Drawing
 PROC. 9TH INTERNATIONAL SYMP. ON ALGORITHMS AND COMPUTATION (ISAAC'98), VOLUME 1533 OF LECTURE NOTES IN COMPUT. SCI
, 1998
"... We study the twolayer planarization problems that have applications in Automatic Graph Drawing. We are searching for a twolayer planar subgraph of maximum weight in a given twolayer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or tw ..."
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Cited by 6 (0 self)
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We study the twolayer planarization problems that have applications in Automatic Graph Drawing. We are searching for a twolayer planar subgraph of maximum weight in a given twolayer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or two, different versions of the problems arise. The latter problem was already investigated in [11] using polyhedral combinatorics. Here, we study
TwoLayer Planarization in Graph Drawing (Extended Abstract)
, 1998
"... ) Abstract We study the twolayer planarization problems that have applications in Automatic Graph Drawing. We are searching for a twolayer planar subgraph of maximum weight in a given twolayer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, o ..."
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) Abstract We study the twolayer planarization problems that have applications in Automatic Graph Drawing. We are searching for a twolayer planar subgraph of maximum weight in a given twolayer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or two, different versions of the problems arise. The latter problem was already investigated in [11] using polyhedral combinatorics. Here, we study the remaining two cases and the relationships between the associated polytopes. In particular, we investigate the polytope P 1 associated with the twolayer planarization problem with one fixed layer. We provide an overview on the relationships between P 1 and the polytope Q 1 associated with the twolayer crossing minimization problem with one fixed layer, the linear ordering polytope, the twolayer planarization problem with zero and two layers fixed. We will see that all facetdefining inequalities in Q 1 are also facetdefining for P 1 ...