To efficiently solve exact discrete optimization problems, branch and bound algorithms require tight bounds. In constraint programming, for optimization, soft arc consistencies typically derive much stronger bounds than those offered by domain or bound consistencies applied to a cost variable. The reason is that soft local consistencies exchange marginal cost information between variables whereas domain consistencies rely only on shrinking domains, which is less informative. However, CP solvers equipped with soft arc consistencies have so far offered limited support for efficient processing of global constraints. In this work, we show how we can efficiently enforce soft local consistency over the AllDifferent constraint, relying on algorithms for the Linear Assignment Problem (LAP). We implement this propagator in toulbar2, the state-of-the-art weighted CP solver exploiting soft local consistencies for bounding. We show that, equipped with this new propagator, toulbar2 outperforms state-of-the-art domain consistency-based CP as well as integer programming solvers for the Quadratic Assignment Problem and shows better performance for miniCOP instances of the 2024 XCSP competition with AllDifferent constraints.

In Constraint Programming, solving discrete minimization problems with hard and soft constraints can be done either using (i) soft global constraints, (ii) a reformulation into a linear program, or (iii) a reformulation into local cost functions. Approach (i) benefits from a vast catalog of constraints. Each soft constraint propagator communicates with other soft constraints only through the variable domains, resulting in weak lower bounds. Conversely, the approach (ii) provides a global view with strong bounds, but the size of the reformulation can be problematic. We focus on approach (iii) in which soft arc consistency (SAC) algorithms produce bounds of intermediate quality. Recently, the introduction of linear constraints as local cost functions increases their modeling expressiveness. We adapt an existing SAC algorithm to handle linear constraints. We show that our algorithm significantly improves the lower bounds compared to the original algorithm on several benchmarks, reducing solving time in some cases.

MOTIVATION: Inferring gene regulatory networks in non-independent genetically related panels is a methodological challenge. This hampers evolutionary and biological studies using heterozygote individuals such as in wild sunflower populations or cultivated hybrids. RESULTS: First, we simulated 100 datasets of gene expressions and polymorphisms, displaying the same gene expression distributions, heterozygosities and heritabilities as in our dataset including 173 genes and 353 genotypes measured in sunflower hybrids. Secondly, we performed a meta-analysis based on six inference methods [least absolute shrinkage and selection operator (Lasso), Random Forests, Bayesian Networks, Markov Random Fields, Ordinary Least Square and fast inference of networks from directed regulation (Findr)] and selected the minimal density networks for better accuracy with 64 edges connecting 79 genes and 0.35 area under precision and recall (AUPR) score on average. We identified that triangles and mutual edges are prone to errors in the inferred networks. Applied on classical datasets without heterozygotes, our strategy produced a 0.65 AUPR score for one dataset of the DREAM5 Systems Genetics Challenge. Finally, we applied our method to an experimental dataset from sunflower hybrids. We successfully inferred a network composed of 105 genes connected by 106 putative regulations with a major connected component. AVAILABILITY AND IMPLEMENTATION: Our inference methodology dedicated to genomic and transcriptomic data is available at https://forgemia.inra.fr/sunrise/inference_methods. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Bayesian networks are probabilistic graphical models with a wide range of application areas including gene regulatory networks inference, risk analysis and image processing. Learning the structure of a Bayesian network (BNSL) from discrete data is known to be an NP-hard task with a superexponential search space of directed acyclic graphs. In this work, we propose a new polynomial time algorithm for discovering a subset of all possible cluster cuts, a greedy algorithm for approximately solving the resulting linear program, and a generalized arc consistency algorithm for the acyclicity constraint. We embed these in the constraint programming-based branch-and-bound solver CPBayes and show that, despite being suboptimal, they improve performance by orders of magnitude. The resulting solver also compares favorably with GOBNILP, a state-of-the-art solver for the BNSL problem which solves an NP-hard problem to discover each cut and solves the linear program exactly.

Graphical models (GMs) define a family of mathematical models aimed at the concise description of multivariate functions using decomposability. We restrict ourselves to functions of discrete variables but try to cover a variety of models that are not always considered as "Graphical Models", ranging from functions with Boolean variables and Boolean co-domain (used in automated reasoning) to functions over finite domain variables and integer or real co-domains (usual in machine learning and statistics). We use a simple algebraic semi-ring based framework for generality, define associated queries, relationships between graphical models, complexity results, and families of algorithms, with their associated guarantees.

Graphical models factorize a global probability distribution/energy function as the product/ sum of local functions. A major inference task, known as MAP in Markov Random Fields and MPE in Bayesian Networks, is to find a global assignment of all the variables with maximum a posteriori probability/minimum energy. A usual distinction on MAP solving methods is complete/incomplete, i.e. the ability to prove optimality or not. Most complete methods rely on tree search, while incomplete methods rely on local search. Among them, we study Variable Neighborhood Search (VNS) for graphical models. In this paper, we propose an iterative approach above VNS that uses (partial) tree search inside its local neighborhood exploration. The proposed approach performs several neighborhood explorations of increasing search complexity, by controlling two parameters, the discrepancy limit and the neighborhood size. Thus, optimality of the obtained solutions can be proven when the neighborhood size is maximal and with unbounded tree search. We further propose a parallel version of our method improving its anytime behavior on difficult instances coming from a large graphical model benchmark. Last we experiment on the challenging minimum energy problem found in Computational Protein Design, showing the practical benefit of our parallel version. A solver is available at https://github.com/toulbar2/toulbar2.

In many applications of constraint programming, it is often impossible to capture all the relevant information in one numerical criterion. In this case, it is useful to produce a set of high quality yet diverse solutions. In this paper, motivated by a Computational Protein Design application, we consider the general problem of producing a diverse set of high-quality solutions of a given Weighted Constraint Satisfaction Problem, with guarantees both on solution quality and diversity. We use weighted automata decomposed in functions of bounded arity, incremental CFN solving, a simple form of predictive bounding and compressed representations of distance constraints for improved efficiency. We show that this approach can be successfully applied to a variety of problems that include both Protein Design Problems but also large Bayesian networks represented as Cost Function Networks. We also show that our approach has the capacity to enumerate so-called local delta-modes and that it does provide improved protein designs.

Motivation: Accurate and economic methods to predict change in protein binding free energy upon mutation are imperative to accelerate the design of proteins for a wide range of applications. Free energy is defined by enthalpic and entropic contributions. Following the recent progresses of Artificial Intelligence-based algorithms for guaranteed NP-hard energy optimization and partition function computation, it becomes possible to quickly compute minimum energy conformations and to reliably estimate the entropic contribution of side-chains in the change of free energy of large protein interfaces. Results: Using guaranteed Cost Function Network algorithms, Rosetta energy functions and Dunbrack's rotamer library, we developed and assessed EasyE and JayZ, two methods for binding affinity estimation that ignore or include conformational entropic contributions on a large benchmark of binding affinity experimental measures. If both approaches outperform most established tools, we observe that side-chain conformational entropy brings little or no improvement on most systems but becomes crucial in some rare cases. Availability and implementation: as open-source Python/C++ code at sourcesup.renater.fr/projects/easy-jayz. Supplementary information: Supplementary data are available at Bioinformatics online.

Virtual Arc Consistency (VAC) is a recent local consistency for processing cost function networks (aka weighted constraint networks) that exploits a simple but powerful connection with standard constraint networks. It has allowed to close hard frequency assignment benchmarks and is capable of directly solving networks of sub modular functions. The algorithm enforcing VAC is an iterative algorithm that solves a sequence of standard constraint networks. This algorithm has been improved by exploiting the idea of dynamic arc consistency between each iteration, leading to the dynamic VAC algorithm. When VAC is maintained during search, the difference between two adjacent nodes in the search tree is also limited. In this paper, we show that the incrementality of Dynamic VAC can also be useful when maintaining VAC during search and we present results showing that maintaining dynamic VAC during search can effectively accelerate search.

MOTIVATION: The main challenge for structure-based computational protein design (CPD) remains the combinatorial nature of the search space. Even in its simplest fixed-backbone formulation, CPD encompasses a computationally difficult NP-hard problem that prevents the exact exploration of complex systems defining large sequence-conformation spaces. RESULTS: We present here a CPD framework, based on cost function network (CFN) solving, a recent exact combinatorial optimization technique, to efficiently handle highly complex combinatorial spaces encountered in various protein design problems. We show that the CFN-based approach is able to solve optimality a variety of complex designs that could often not be solved using a usual CPD-dedicated tool or state-of-the-art exact operations research tools. Beyond the identification of the optimal solution, the global minimum-energy conformation, the CFN-based method is also able to quickly enumerate large ensembles of suboptimal solutions of interest to rationally build experimental enzyme mutant libraries. AVAILABILITY: The combined pipeline used to generate energetic models (based on a patched version of the open source solver Osprey 2.0), the conversion to CFN models (based on Perl scripts) and CFN solving (based on the open source solver toulbar2) are all available at http://genoweb.toulouse.inra.fr/~tschiex/CPD

As (Lee et al., 2012) have shown, weighted constraint satisfaction problems can benefit from the introduction of global cost functions, leading to a new Cost Function Programming paradigm. In this paper, we explore the possibility of decomposing global cost functions in such a way that enforcing soft local consistencies on the decomposition offers guarantees on the level of consistency enforced on the original global cost function. We show that directional arc consistency and virtual arc consistency offer such guarantees. We conclude by experiments on decomposable cost functions showing that decompositions may be very useful to easily integrate efficient global cost functions in solvers.

MOTIVATION: Genome maps are imperative to address the genetic basis of the biology of an organism. While a growing number of genomes are being sequenced providing the ultimate genome maps-this being done at an even faster pace now using new generation sequencers-the process of constructing intermediate maps to build and validate a genome assembly remains an important component for producing complete genome sequences. However, current mapping approach lack statistical confidence measures necessary to identify precisely relevant inconsistencies between a genome map and an assembly. RESULTS: We propose new methods to derive statistical measures of confidence on genome maps using a comparative model for radiation hybrid data. We describe algorithms allowing to (i) sample from a distribution of maps and (ii) exploit this distribution to construct robust maps. We provide an example of application of these methods on a dog dataset that demonstrates the interest of our approach. AVAILABILITY: Methods are implemented in two freely available softwares: Carthagene (http://www.inra.fr/mia/T/CarthaGene/) and a companion software (metamap, available at: http://snp.toulouse.inra.fr/~servin/index.cgi/Metamap).

The Weighted Constraint Satisfaction Problem (WCSP) framework allows representing and solving problems involving both hard constraints and cost functions. It has been applied to various problems, including resource allocation, bioinformatics, scheduling, etc. To solve such problems, solvers usually rely on branch-and-bound algorithms equipped with local consistency filtering, mostly soft arc consistency. However, these techniques are not well suited to solve problems with very large domains. Motivated by the resolution of an RNA gene localization problem inside large genomic sequences, and in the spirit of bounds consistency for large domains in crisp CSPs, we introduce soft bounds arc consistency, a new weighted local consistency specifically designed for WCSP with very large domains. Compared to soft arc consistency, BAC provides significantly improved time and space asymptotic complexity. In this paper, we show how the semantics of cost functions can be exploited to further improve the time complexity of BAC. We also compare both in theory and in practice the efficiency of BAC on a WCSP with bounds consistency enforced on a crisp CSP using cost variables. On two different real problems modeled as WCSP, including our RNA gene localization problem, we observe that maintaining bounds arc consistency outperforms arc consistency and also improves over bounds consistency enforced on a constraint model with cost variables.

MOTIVATION: Genome maps are fundamental to the study of an organism and essential in the process of genome sequencing which in turn provides the ultimate map of the genome. The increased number of genomes being sequenced offers new opportunities for the mapping of closely related organisms. We propose here an algorithmic formalization of a genome comparison approach to marker ordering. RESULTS: In order to integrate a comparative mapping approach in the algorithmic process of map construction and selection, we propose to extend the usual statistical model describing the experimental data, here radiation hybrids (RH) data, in a statistical framework that models additionally the evolutionary relationships between a proposed map and a reference map: an existing map of the corresponding orthologous genes or markers in a closely related organism. This has concretely the effect of exploiting, in the process of map selection, the information of marker adjacencies in the related genome when the information provided by the experimental data is not conclusive for the purpose of ordering. In order to compute efficiently the map, we proceed to a reduction of the maximum likelihood estimation to the Traveling Salesman Problem. Experiments on simulated RH datasets as well as on a real RH dataset from the canine RH project show that maps produced using the likelihood defined by the new model are significantly better than maps built using the traditional RH model. AVAILABILITY: The comparative mapping approach is available in the last version of de Givry,S. et al. [(2004) Bioinformatics, 21, 1703-1704, www.inra.fr/mia/T/CarthaGene], a free (the LKH part is free for academic use only) mapping software in C++, including LKH (Helsgaun,K. (2000) Eur. J. Oper. Res., 126, 106-130, www.dat.ruc.dk/keld/research/LKH) for maximum likelihood computation.

MOTIVATION: Searching RNA gene occurrences in genomic sequences is a task whose importance has been renewed by the recent discovery of numerous functional RNA, often interacting with other ligands. Even if several programs exist for RNA motif search, none exists that can represent and solve the problem of searching for occurrences of RNA motifs in interaction with other molecules. RESULTS: We present a constraint network formulation of this problem. RNA are represented as structured motifs that can occur on more than one sequence and which are related together by possible hybridization. The implemented tool MilPat is used to search for several sRNA families in genomic sequences. Results show that MilPat allows to efficiently search for interacting motifs in large genomic sequences and offers a simple and extensible framework to solve such problems. New and known sRNA are identified as H/ACA candidates in Methanocaldococcus jannaschii. AVAILABILITY: http://carlit.toulouse.inra.fr/MilPaT/MilPat.pl.

UNLABELLED: CAR(H)(T)A GENE: is an integrated genetic and radiation hybrid (RH) mapping tool which can deal with multiple populations, including mixtures of genetic and RH data. CAR(H)(T)A GENE: performs multipoint maximum likelihood estimations with accelerated expectation-maximization algorithms for some pedigrees and has sophisticated algorithms for marker ordering. Dedicated heuristics for framework mapping are also included. CAR(H)(T)A GENE: can be used as a C++ library, through a shell command and a graphical interface. The XML output for companion tools is integrated. AVAILABILITY: The program is available free of charge from www.inra.fr/bia/T/CarthaGene for Linux, Windows and Solaris machines (with Open Source). CONTACT: [email protected].