Arc-annotated sequences are useful in representing the structural information of RNA sequences.

. Arc-annotated sequences are useful in representing the structural information of RNA and protein sequences. Recently, the longest arc-preserving common subsequence problem has been introduced in [6, 7] as a framework for studying the similarity of arc-annotated sequences. In this paper, we consider arc-annotated sequences with various arc structures and present some new algorithmic and complexity results on the longest arc-preserving common subsequence problem. Some of our results answer an open question in [6, 7] and some others improve the hardness results in [6, 7]. Keywords: sequence annotation, longest common subsequence, approximation algorithm, maximum independent set, MAX SNP-hard, dynamic programming. 1

CATEGORY: Real-World Applications Structural comparison of proteins is a core problem in modern biomedical research.

This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations, structure prediction and recognition. Additional biology is given in the context of the problems, including some motivation for disease diagnosis and drug discovery. Open problems are cited with an extensive bibliography, and we o er a guide to getting started in this exciting frontier.

In this paper we describe the application of a so called "Self-Generating" Memetic Algorithm to the Maximum Contact Map Overlap problem (MAX-CMO). The maximum overlap of contact maps is emerging as a leading modeling technique to obtain structural alignment among pairs of protein structures. Identifying structural alignments (and hence similarity among proteins) is essential to the correct assessment of the relation between proteins structure and function. A robust methodology for structural comparison could have impact on the process of rational drug design.

We present exact algorithms for the NP-complete Longest Common Subsequence problem for sequences with nested arc annotations, a problem occurring in structure comparison of RNA. Given two sequences of length at most n and nested arc structure, one of our algorithms determines (if existent) in O(3.31 time an arc-preserving subsequence of both sequences, which can be obtained by deleting (together with corresponding arcs) k 1 letters from the first and k 2 letters from the second sequence. A second algorithm shows that (in case of a four letter alphabet) we can find a length l arc-annotated subsequence in O(12 n) time. This means that the problem is fixed-parameter tractable when parameterized by the number of deletions as well as when parameterized by the subsequence length. Our findings complement known approximation results which give a quadratic time factor-2-approximation for the general and polynomial time approximation schemes for restricted versions of the problem. In addition, we obtain further fixed-parameter tractability results for these restricted versions.

Many important problems arising in computational biochemistry and genomics have been formulated in terms of underlying combinatorial optimization models. In particular, a number have been formulated as clique-detection models. The proposed article includes an introduction to the underlying biochemistry and genomic aspects of the problems as well as to the graph-theoretic aspects of the solution approaches. Each subsequent section describes a particular type of problem, gives an example to show how the graph model can be derived, summarizes recent progress, and discusses challenges associated with solving the associated graph-theoretic models. Clique detection models include prescribing (a) a maximal clique, (b) a maximum clique, (c) a maximum weighted clique, or (d) all maximal cliques in a graph. The particular types of biochemistry and genomics problems that can be represented by a clique detection model include integration of genome mapping data, nonoverlapping local alignments, matching and comparing molecular structures, and protein docking.

A vast number of very successful applications of Global-Local Search Hybrids have been reported in the literature in the last years for a wide range of problem domains. The majority of these papers report the combination of highly specialized pre-existing local searchers and usually purpose-speci c global operators (e.g. genetic operators in an Evolutionary Algorithm).

Abstract. We address the problem of approximating the 2-Interval Pattern problem over its various models and restrictions. This problem, which is motivated by RNA secondary structure prediction, asks to find a maximum cardinality subset of a 2-interval set with respect to some prespecified model. For each such model, we give varying approximation quality depending on the different possible restrictions imposed on the input 2-interval set. 1

Abstract. In molecular biology, RNA structure comparison and motif search are of great interest for solving major problems such as phylogeny reconstruction, prediction of molecule folding and identification of common functions. RNA structures can be represented by arc-annotated sequences (primary sequence along with arc annotations), and this paper mainly focuses on the so-called arc-preserving subsequence (APS) problem where, given two arc-annotated sequences (S, P) and (T, Q), we are asking whether (T, Q) can be obtained from (S, P) by deleting some of its bases (together with their incident arcs, if any). In previous studies, this problem has been naturally divided into subproblems reflecting the intrinsic complexity of the arc structures. We show that APS(Crossing, Plain) is NP-complete, thereby answering an open problem posed in [11]. Furthermore, to get more insight into where the actual border between the polynomial and the NP-complete cases lies, we refine the classical subproblems of the APS problem in much the same way as in [19] and prove that both APS({⊏, ≬}, ∅) and APS({<, ≬}, ∅) are NPcomplete. We end this paper by giving some new positive results, namely showing that APS({≬}, ∅) and APS({≬},{≬}) are polynomial time solvable. Keywords: RNA structures, Arc-Preserving Subsequence problem, Computational complexity.