Absolute Convergence of Rational Series is Semi-decidable

International audienceWe study \emph{real-valued absolutely convergent rational series}, i.e. functions $r: \Sigma^* \rightarrow {\mathbb R}$, defined over a free monoid $\Sigma^*$, that can be computed by a multiplicity automaton $A$ and such that $\sum_{w\in \Sigma^*}|r(w)|<\infty$. We prove that any absolutely convergent rational series $r$ can be computed by a multiplicity automaton $A$ which has the property that $r_{|A|}$ is simply convergent, where $r_{|A|}$ is the series computed by the automaton $|A|$ derived from $A$ by taking the absolute values of all its parameters. Then, we prove that the set ${\cal A}^{rat}(\Sigma)$ composed of all absolutely convergent rational series is semi-decidable and we show that the sum $\sum_{w\in \Sigma^*}|r(w)|$ can be estimated to any accuracy rate for any $r\in {\cal A}^{rat}(\Sigma)$. We also introduce a spectral radius-like parameter $\rho_{|r|}$ which satisfies the following property: $r$ is absolutely convergent iff $\rho_{|r|}<1$

Inapproximability of maximal strip recovery

In comparative genomic, the first step of sequence analysis is usually to decompose two or more genomes into syntenic blocks that are segments of homologous chromosomes. For the reliable recovery of syntenic blocks, noise and ambiguities in the genomic maps need to be removed first. Maximal Strip Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff for reliably recovering syntenic blocks from genomic maps in the midst of noise and ambiguities. Given $d$ genomic maps as sequences of gene markers, the objective of \msr{d} is to find $d$ subsequences, one subsequence of each genomic map, such that the total length of syntenic blocks in these subsequences is maximized. For any constant $d \ge 2$, a polynomial-time 2d-approximation for \msr{d} was previously known. In this paper, we show that for any $d \ge 2$, \msr{d} is APX-hard, even for the most basic version of the problem in which all gene markers are distinct and appear in positive orientation in each genomic map. Moreover, we provide the first explicit lower bounds on approximating \msr{d} for all $d \ge 2$. In particular, we show that \msr{d} is NP-hard to approximate within $\Omega(d/\log d)$. From the other direction, we show that the previous 2d-approximation for \msr{d} can be optimized into a polynomial-time algorithm even if $d$ is not a constant but is part of the input. We then extend our inapproximability results to several related problems including \cmsr{d}, \gapmsr{\delta}{d}, and \gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009) and the Proceedings of the 4th International Frontiers of Algorithmics Workshop (FAW 2010

Impact Of The Energy Model On The Complexity Of RNA Folding With Pseudoknots

International audiencePredicting the folding of an RNA sequence, while allowing general pseudoknots (PK), consists in finding a minimal free-energy matching of its $n$ positions. Assuming independently contributing base-pairs, the problem can be solved in $\Theta(n^3)$-time using a variant of the maximal weighted matching. By contrast, the problem was previously proven NP-Hard in the more realistic nearest-neighbor energy model. In this work, we consider an intermediate model, called the stacking-pairs energy model. We extend a result by Lyngs\o, showing that RNA folding with PK is NP-Hard within a large class of parametrization for the model. We also show the approximability of the problem, by giving a practical $\Theta(n^3)$ algorithm that achieves at least a $5$-approximation for any parametrization of the stacking model. This contrasts nicely with the nearest-neighbor version of the problem, which we prove cannot be approximated within any positive ratio, unless $P=NP$.La prédiction du repliement, avec pseudonoeuds généraux, d'une séquence d'ARN de taille $n$ est équivalent à la recherche d'un couplage d'énergie libre minimale. Dans un modèle d'énergie simple, où chaque paire de base contribue indépendamment à l'énergie, ce problème peut être résolu en temps $\Theta(n^3)$ grâce à une variante d'un algorithme de couplage pondéré maximal. Cependant, le même problème a été démontré NP-difficile dans le modèle d'énergie dit des plus proches voisins. Dans ce travail, nous étudions les propriétés du problème sous un modèle d'empilements, constituant un modèle intermédiaire entre ceux d'appariement et des plus proches voisins. Nous démontrons tout d'abord que le repliement avec pseudo-noeuds de l'ARN reste NP-difficile dans de nombreuses valuations du modèle d'énergie. . Par ailleurs, nous montrons que ce problème est approximable, en proposant un algorithme polynomial garantissant une $1/5$-approximation. Ce résultat illustre une différence essentielle entre ce modèle et celui des plus proches voisins, pour lequel nous montrons qu'il ne peut être approché à aucun ratio positif par un algorithme en temps polynomial sauf si $N=NP$

A Combinatorial Framework for Designing (Pseudoknotted) RNA Algorithms

We extend an hypergraph representation, introduced by Finkelstein and Roytberg, to unify dynamic programming algorithms in the context of RNA folding with pseudoknots. Classic applications of RNA dynamic programming energy minimization, partition function, base-pair probabilities...) are reformulated within this framework, giving rise to very simple algorithms. This reformulation allows one to conceptually detach the conformation space/energy model -- captured by the hypergraph model -- from the specific application, assuming unambiguity of the decomposition. To ensure the latter property, we propose a new combinatorial methodology based on generating functions. We extend the set of generic applications by proposing an exact algorithm for extracting generalized moments in weighted distribution, generalizing a prior contribution by Miklos and al. Finally, we illustrate our full-fledged programme on three exemplary conformation spaces (secondary structures, Akutsu's simple type pseudoknots and kissing hairpins). This readily gives sets of algorithms that are either novel or have complexity comparable to classic implementations for minimization and Boltzmann ensemble applications of dynamic programming

Codesigning a Measure of Person-Centred Coordinated Care to Capture the Experience of the Patient: The Development of the P3CEQ

Background: Person-centred coordinated care (P3C) is a priority for stakeholders (ie, patients, carers, professionals, policy makers). As a part of the development of an evaluation framework for P3C, we set out to identify patient-reported experience measures (PREMs) suitable for routine measurement and feedback during the development of services. Methods: A rapid review of the literature was undertaken to identity existing PREMs suitable for the probing person-centred and/or coordinated care. Of 74 measures identified, 7 met our inclusion criteria. We critically examined these against core domains and subdomains of P3C. Measures were then presented to stakeholders in codesign workshops to explore acceptability, utility, and their strengths/weaknesses. Results: The Long-Term Condition 6 questionnaire was preferred for its short length, utility, and tone. However, it lacked key questions in each core domain, and in response to requests from our codesign group, new questions were added to cover consideration as a whole person, coordination, care plans, carer involvement, and a single coordinator. Cognitive interviews, on-going codesign, and mapping to core P3C domains resulted in the refinement of the questionnaire to 11 items with 1 trigger question. The 11-item modified version was renamed the P3C Experiences Questionnaire. Conclusions: Due to a dearth of brief measures available to capture people’s experience of P3C for routine practice, an existing measure was modified using an iterative process of adaption and validation through codesign workshops. Next steps include psychometric validation and modification for people with dementia and learning difficulties.</p