Ribonucleic acid (RNA) is a polymer composed of four bases denoted A, C, G, and U. It generally is a single-stranded 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...

The Train Timetabling Problem aims at determining a periodic timetable for a set of trains which does not violate track capacities and satises some operational constraints. In particular, we concentrate on the problem of a single, one-way track linking two major stations, with a number of intermediate stations in between. Each train connects two given stations along the track (possibly dierent from the two major stations) and may have to stop for a minimum time in some of the intermediate stations. Trains can overtake each other only in correspondence of an intermediate station, and a minimum time interval between two consecutive departures and arrivals of trains in each station is specied. In this paper, we propose a graph theoretic formulation for the problem using a directed multigraph in which nodes correspond to departures/arrivals at a certain station at a given time instant. This formulation is used to derive an integer linear programming model which is relaxed in a Lagrangian way. A novel feature of our model is that the variables in the relaxed constraints are associated only with nodes (as opposed to arcs) of the aforementioned graph. This allows a considerable speedup in the solution of the relaxation. The relaxation is embedded within a heuristic algorithm which makes extensive use of the dual information associated with the Lagrangian multipliers. We report extensive computational results on real-world instances provided from Ferrovie dello Stato SpA, the Italian railway company, and from Ansaldo Segnalamento Ferroviario SpA.

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The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an object-oriented software framework for branch-and-cut-and-price algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branch-and-cut-and-price algorithms for combinatorial optimization problems and shows how they are managed by ABACUS.

Multiple sequence alignment is a central problem in Bioinformatics and a challenging one for optimisation algorithms. An established integer programming approach is to apply branch-and-cut to a graph-theoretical model. The models are exponentially large but are represented intensionally, and violated constraints can be located in polynomial time. This report describes a new integer program formulation that generates polynomial-sized models small enough to be passed to generic solvers. It is a hybrid formulation relating the sparse alignment graph with a compact encoding of the alignment matrix via channelling constraints. Alignments obtained with a pseudo-Boolean local search algorithm are competitive with those of state-of-the-art algorithms. Execution times are much longer, but in future work we aim to develop a more efficient specialised algorithm. 1

Abstract. Multiple sequence alignment is a central problem in Bioinformatics. A known integer programming approach is to apply branch-and-cut to exponentially large graph-theoretic models. This paper describes a new integer program formulation that generates models small enough to be passed to generic solvers. The formulation is a hybrid relating the sparse alignment graph with a compact encoding of the alignment matrix via channelling constraints. Alignments obtained with a SAT-based local search algorithm are competitive with those of state-of-the-art algorithms, though execution times are much longer. 1

Abstract. Computational alignment of a biopolymer sequence (e.g., an RNA or a protein) to a structure is an effective approach to predict and search for the structure of new sequences. To identify the structure of remote homologs, the structure-sequence alignment has to consider not only sequence similarity but also spatially conserved conformations caused by residue interactions, and consequently is computationally intractable. It is difficult to cope with the inefficiency without compromising alignment accuracy, especially for structure search in genomes or large databases. This paper introduces a novel method and a parameterized algorithm for structuresequence alignment. Both the structure and the sequence are represented as graphs, where in general the graph for a biopolymer structure has a naturally small tree width. The algorithm constructs an optimal alignment by finding in the sequence graph the maximum valued subgraph isomorphic to the structure graph. It has the computational time complexity O(k t N 2) for the structure of N residues and its tree decomposition of width t. The parameter k, small in nature, is determined by a statistical cutoff for the correspondence between the structure and the sequence. The paper demonstrates a successful application of the algorithm to developing a fast program for RNA structural homology search. 1

We study the two-layer planarization problems that have applications in Automatic Graph Drawing. We are searching for a two-layer planar subgraph of maximum weight in a given two-layer 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

Extracting multiple structural alignments from pairwise alignments: a comparison of a rigorous and a heuristic approach

) Abstract We study the two-layer planarization problems that have applications in Automatic Graph Drawing. We are searching for a two-layer planar subgraph of maximum weight in a given two-layer 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 two-layer planarization problem with one fixed layer. We provide an overview on the relationships between P 1 and the polytope Q 1 associated with the two-layer crossing minimization problem with one fixed layer, the linear ordering polytope, the two-layer planarization problem with zero and two layers fixed. We will see that all facet-defining inequalities in Q 1 are also facet-defining for P 1 ...