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We generalize a problem of finding maximum-scoring segment sets, previously studied by Csurös (IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2004, 1, 139-150), from sequences to graphs. Namely, given a vertex-weighted graph G and a non-negative startup penalty c, we can find a set of vertex-disjoint paths in G with maximum total score when each path's score is its vertices' total weight minus c. We call this new problem maximum-scoring path sets (MSPS). We present an algorithm that has a linear-time complexity for graphs with a constant treewidth. Generalization from sequences to graphs allows the algorithm to be used on pangenome graphs representing several related genomes and can be seen as a common abstraction for several biological problems on pangenomes, including searching for CpG islands, ChIP-seq data analysis, analysis of region enrichment for functional elements, or simple chaining problems.
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Although RNA secondary structure prediction is a textbook application of dynamic programming (DP) and routine task in RNA structure analysis, it remains challenging whenever pseudoknots come into play. Since the prediction of pseudoknotted structures by minimizing (realistically modelled) energy is NP-hard, specialized algorithms have been proposed for restricted conformation classes that capture the most frequently observed configurations. To achieve good performance, these methods rely on specific and carefully hand-crafted DP schemes. In contrast, we generalize and fully automatize the design of DP pseudoknot prediction algorithms. For this purpose, we formalize the problem of designing DP algorithms for an (infinite) class of conformations, modeled by (a finite number of) fatgraphs, and automatically build DP schemes minimizing their algorithmic complexity. We propose an algorithm for the problem, based on the tree-decomposition of a well-chosen representative structure, which we simplify and reinterpret as a DP scheme. The algorithm is fixed-parameter tractable for the treewidth tw of the fatgraph, and its output represents a [Formula: see text] algorithm (and even possibly [Formula: see text] in simple energy models) for predicting the MFE folding of an RNA of length n. We demonstrate, for the most common pseudoknot classes, that our automatically generated algorithms achieve the same complexities as reported in the literature for hand-crafted schemes. Our framework supports general energy models, partition function computations, recursive substructures and partial folding, and could pave the way for algebraic dynamic programming beyond the context-free case.
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BACKGROUND: Phylogenetic reconstruction is one of the paramount challenges of contemporary bioinformatics. A subtask of existing tree reconstruction algorithms is modeled by the SMALL PARSIMONY problem: given a tree T and an assignment of character-states to its leaves, assign states to the internal nodes of T such as to minimize the parsimony score, that is, the number of edges of T connecting nodes with different states. While this problem is polynomial-time solvable on trees, the matter is more complicated if T contains reticulate events such as hybridizations or recombinations, i.e. when T is a network. Indeed, three different versions of the parsimony score on networks have been proposed and each of them is NP-hard to decide. Existing parameterized algorithms focus on combining the number c of possible character-states with the number of reticulate events (per biconnected component). RESULTS: We consider the parameter treewidth t of the underlying undirected graph of the input network, presenting dynamic programming algorithms for (slight generalizations of) all three versions of the parsimony problem on size-n networks running in times [Formula: see text], [Formula: see text], and [Formula: see text], respectively. Our algorithms use a formulation of the treewidth that may facilitate formalizing treewidth-based dynamic programming algorithms on phylogenetic networks for other problems. CONCLUSIONS: Our algorithms allow the computation of the three popular parsimony scores, modeling the evolutionary development of a (multistate) character on a given phylogenetic network of low treewidth. Our results subsume and improve previously known algorithm for all three variants. While our results rely on being given a "good" tree-decomposition of the input, encouraging theoretical results as well as practical implementations producing them are publicly available. We present a reformulation of tree decompositions in terms of "agreeing trees" on the same set of nodes. As this formulation may come more natural to researchers and engineers developing algorithms for phylogenetic networks, we hope to render exploiting the input network's treewidth as parameter more accessible to this audience.
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Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed-Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the TREE-DIET problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth [Formula: see text]. Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account. Our core result is an FPT dynamic programming algorithm for TREE-DIET, using [Formula: see text] time and space. We complement this result with parameterized complexity lower-bounds for stronger variants (e.g., NP-hardness when [Formula: see text] or [Formula: see text] is constant). We propose a prototype implementation for our approach which we apply on difficult instances of selected RNA-based problems: RNA design, sequence-structure alignment, and search of pseudoknotted RNAs in genomes, revealing very encouraging results. This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics.
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We consider the # P -complete problem of counting the number of linear extensions of a poset ( # LE ) ; a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of # LE parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that # LE is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that # LE becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.
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A secure set S in a graph is defined as a set of vertices such that for any XâS the majority of vertices in the neighborhood of X belongs to S. It is known that deciding whether a set S is secure in a graph is co-NP -complete. However, it is still open how this result contributes to the actual complexity of deciding whether for a given graph G and integer k, a non-empty secure set for G of size at most k exists. In this work, we pinpoint the complexity of this problem by showing that it is Σ2P -complete. Furthermore, the problem has so far not been subject to a parameterized complexity analysis that considers structural parameters. In the present work, we prove that the problem is W[1] -hard when parameterized by treewidth. This is surprising since the problem is known to be FPT when parameterized by solution size and "subset problems" that satisfy this property usually tend to be FPT for bounded treewidth as well. Finally, we give an upper bound by showing membership in XP , and we provide a positive result in the form of an FPT algorithm for checking whether a given set is secure on graphs of bounded treewidth.
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A graph H is a square root of a graph G, or equivalently, G is the square of H, if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. The problem of testing whether a graph admits a square root which belongs to some specified graph class â is called the â -Square Root problem. By showing boundedness of treewidth we prove that Square Root is polynomial-time solvable on some classes of graphs with small clique number and that â -Square Root is polynomial-time solvable when â is the class of cactuses.