The Quadratic Assignment Problem (QAP) consists of finding a permutation that minimizes a quadratic objective function. Exact methods generally rely on a branch-and-bound procedure, the efficiency of which depends heavily on the quality of its lower bound. In integer linear programming, several bounds have been investigated, exhibiting different trade-offs between speed and quality. The Gilmore-Lawler bound appears to be the most commonly used in practice. It requires solving a linear assignment problem (LAP) for each variable-value pair. We show how to obtain this bound using Singleton Node Consistency (SNC) and LAP. In Cost Function Networks (CFNs), we propose a reformulation that transforms the result of applying LAP to a given variable-value pair into cost functions of arity 1 and 2, which can be added to the original problem. Combined with existing lower bounds for CFNs, including EDAC and a recent CFN propagator for AllDifferent, this method (SNC-LAP-GLB), used as a preprocessing, significantly increases the initial lower bound and accelerates the search, resulting in competitive results on the QAPLIB benchmark. We then propose an extension of the AllDifferent propagator for the Global Cardinality Constraint. It allows us to exploit variable symmetries on some challenging QAPLIB instances, thus improving the results.

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.

The optimal allocation of crops to different parcels of land is a problem of paramount practical importance, not only to improve food and feed production, but also to address the challenges posed by climate change. However, this optimization problem is inherently complex due to the large number of agricultural sites available which generates a vast search space that renders traditional optimization techniques impractical. Moreover, as maximizing average production may generate solutions characterized by high year-by-year instability and lead to large and unrealistic cultivated areas, it is necessary to optimize crop allocation considering several objectives at the same time. In order to tackle this complex optimization problem, we propose a multi-objective approach, simultaneously maximizing the average production, minimizing the year-on-year production variance, and minimizing the total cultivated surface. The approach relies on an established multi-objective evolutionary algorithm, and employs a machine learning model able to predict crop production from weather and irrigation conditions, trained on historical data, making it possible to tackle allocation problems of large size. The proposed approach is compared to a quadratic programming algorithm tailored to the target problem. A case study focusing on the allocation of soybean crops in the European continent for the years 2000-2023 shows that the proposed methodology is able to identify informative trade-offs between the three conflicting objectives considered, and identify realistic and meaningful crop allocations for supporting stakeholders’ decisions.

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.

The core-guided algorithm OLL is the basis of some of the most successful algorithms for MaxSAT in recent evaluations. It works by iteratively finding cores of the formula and transforming it so that it exhibits a higher lower bound. It has recently been shown to implicitly discover cores of the original formula, as well as a compact representation of its reasoning within a linear program. In this paper, we use and extend these results to design a practical MaxSAT solver. We show an explicit linear program which matches and usually exceeds the bound computed by OLL. We show that OLL can be restated as an algorithm that explicitly computes a feasible dual solution of this linear program. This restated algorithm naturally works with an arbitrary dual solution. It can in fact be used to improve any LP representation of the MaxSAT instance. This presents a large increase of the potential design space for such algorithms. We describe some potential improvements from this insight and show that an implementation outperforms the state of the art algorithms on the set of instances from the latest MaxSAT evaluation.

In constraint satisfaction, pairwise consistency (PWC) is a well-known local consistency improving generalized arc consistency in theory but not often in practice. A popular approach to enforcing PWC enforces arc consistency on the dual encoding of the problem, allowing to reuse existing AC algorithms. In this paper, we explore the benefit of this simple approach in the optimization context of cost function networks and soft local consistencies. Using a dual encoding, we obtain an equivalent binary cost function network where enforcing virtual arc consistency achieves virtual PWC on the original problem. We experimentally observed that adding extra non-binary cost functions before the dual encoding results in even stronger bounds. Such supplementary cost functions may be produced by bounded variable elimination or by adding ternary zero-cost functions. Experiments on (probabilistic) graphical models, from the UAI 2022 competition benchmark, show a clear improvement when using our approach inside a branch-and-bound solver compared to the state-of-the-art.

Learning constraint networks is known to require a number of membership queries exponential in the number of variables. In this paper, we learn constraint networks by asking the user partial queries. That is, we ask the user to classify assignments to subsets of the variables as positive or negative. We provide an algorithm, called QuAcq2, that, given a negative example, elucidates a constraint of the target network in a number of queries logarithmic in the size of the example. The whole constraint network can then be learned with a polynomial number of partial queries. We give information theoretic lower bounds for learning some simple classes of constraint networks and show that our generic algorithm is optimal in some cases. We provide a version of QuAcq2 with a cutoff mechanism that controls the time to generate a query. Our experiments illustrate the good behavior of QuAcq2 in practice, especially in the case where QuAcq2 is executed to learn the missing constraints in a partially filled constraint model. Our experiments also show that QuAcq2 requires significantly fewer queries to learn a network than its predecessor QuAcq1.

Many current complete MaxSAT algorithms fall into two categories: core-guided or implicit hitting set. The two kinds of algorithms seem to have complementary strengths in practice, so that each kind of solver is better able to handle different families of instances. This suggests that a hybrid might match and outperform either, but the techniques used seem incompatible. In this paper, we focus on PMRES and OLL, two core-guided algorithms based on max resolution and soft cardinality constraints, respectively. We show that these algorithms implicitly discover cores of the original formula, as has been previously shown for PM1. Moreover, we show that in some cases, including unweighted instances, they compute the optimum hitting set of these cores at each iteration. We also give compact integer linear programs for each which encode this hitting set problem. Importantly, their continuous relaxation has an optimum that matches the bound computed by the respective algorithms. This goes some way towards resolving the incompatibility of implicit hitting set and core-guided algorithms, since solvers based on the implicit hitting set algorithm typically solve the problem by encoding it as a linear program.

Graphical models, such as cost function networks (CFNs), can compactly express large decomposable functions, which leads to efficient inference algorithms. Most methods for computing lower bounds in Branch-and-Bound minimization compute feasible dual solutions of a specific linear relaxation. These methods are more effective than solving the linear relaxation exactly, with better worst-case time complexity and better performance in practice. However, these algorithms are specialized to the structure of the linear relaxation of a CFN and cannot, for example, deal with constraints that cannot be expressed in extension, such as linear constraints of large arity. In this work, we show how to extend soft local consistencies, a set of approximate inference techniques for CFNs, so that they handle linear constraints, as well as combinations of linear constraints with at-most-one constraints. We embedded the resulting algorithm in toulbar2, an exact Branch-and-Bound solver for CFNs which has demonstrated superior results in several graphical model competitions and is state-of-the-art for solving large computational protein design (CPD) problems. We significantly improved performance of the solver in CPD with diversity guarantees. It also compared favorably with integer linear programming solvers on knapsack problems with conflict graphs.

While processor frequency has stagnated over the past two decades, the number of available cores in servers or clusters is still growing, offering the opportunity for significant speed-up in combinatorial optimization. Parallelization of exact methods remains a difficult challenge. We revisit the concept of parallel Branch-and-Bound in the framework of Cost Function Networks. We show how to adapt the anytime Hybrid Best-First Search algorithm in a Master-Worker protocol. The resulting parallel algorithm achieves good load-balancing without introducing new parameters to be tuned as is the case, for example, in Embarrassingly Parallel Search (EPS). It has also a small overhead due to its light communication messages. We performed an experimental evaluation on several benchmarks, comparing our parallel algorithm to its sequential version. We observed linear speed-up in some cases. Our approach compared favourably to the EPS approach and also to a state-of-the-art parallel exact integer programming solver.

In this paper, we extend a Burer-Monteiro style method to compute low rank Semi-Definite Programming (SDP) bounds for the MAP problem on discrete graphical models with an arbitrary number of states and arbitrary pairwise potentials. We consider both a penalized constraint approach and a dedicated Block Coordinate Descent (BCD) approach which avoids large penalty coefficients in the cost matrix. We show our algorithm is decreas-ing. Experiments show that the BCD approach compares favorably to the penalized approach and to usual linear bounds relying on convergent message passing approaches.

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.

Towards the goal of explainable AI, Bayesian networks offer a rich framework for probabilistic reasoning. Bayesian Network Structure Learning (BNSL) from discrete observations corresponds to finding a compact model which best explains the data. It defines an NP-hard problem with a superexponential search space of Directed Acyclic Graphs (DAG). Several constraint-based (exploiting local conditional independence tests) and score-based (exploiting a global objective formulation) BNSL methods have been developed in the past. Complete methods for score-based BNSL include

Constraint programming is a state of the art technique for learning the structure of Bayesian Networks from data (Bayesian Network Structure Learning – BNSL). However, scalability both for CP and other combinatorial optimization techniques for this problem is limited by the fact that the basic decision variables are set variables with domain sizes that may grow super polynomially with the number of random variables. Usual techniques for handling set variables in CP are not useful, as they lead to poor bounds. In this paper, we propose using decision trees as a data structure for storing sets of sets to represent set variable domains. We show that relatively simple operations are sufficient to implement all propagation and bounding algorithms, and that the use of these data structures improves scalability of a state of the art CP-based solver for BNSL. Computing methodologies → Learning in probabilistic graphical models; Theory of computation → Discrete optimization

∗Papier doctorant : Pierre Montalbano1 est auteur principal. Cost function networks (CFNs), also known as Weighted Constraint Satisfaction Problems, can compactly express large decomposable cost functions, which leads to efficient inference algorithms. In particular, most methods for computing dual (lower) bounds in minimization compute feasible dual solutions of a specific linear relaxation. These methods are more effective than solving the linear relaxation exacly, because its size is significantly larger than the original CFN. However, these algorithms are specialized to the structure of the linear relaxation of a CFN and cannot, for example, deal with constraints that cannot be expressed in extension, such as linear constraints of large arity. In this work, we show how to extend soft local consistencies, a set of approximate inference techniques for CFNs, so that they handle linear constraints, as well as combinations of linear constraints with at-most-one constraints. We embedded the resulting algorithm in toulbar2, an exact Branch-and-Bound solver for CFNs which primarily relies on EDAC for dual bounds during search. This allows us to test this solver on problems on which it was previously not applicable, such as pseudoBoolean optimization (PB) and knapsack with conflict graph (KPCG). Compared to dedicated PB solvers and integer linear programming solvers, we demonstrate stateof-the-art performance on some families from recent PB optimization evaluations and KPCG, and improve performance in computational protein design with diversity

Proteins are the main active molecules of life. Although natural proteins play many roles, as enzymes or antibodies for example, there is a need to go beyond the repertoire of natural proteins to produce engineered proteins that precisely meet application requirements, in terms of function, stability, activity or other protein capacities. Computational Protein Design aims at designing new proteins from first principles, using full-atom molecular models. However, the size and complexity of proteins require approximations to make them amenable to energetic optimization queries. These approximations make the design process less reliable, and a provable optimal solution may fail. In practice, expensive libraries of solutions are therefore generated and tested. In this paper, we explore the idea of generating libraries of provably diverse low-energy solutions by extending cost function network algorithms with dedicated automaton-based diversity constraints on a large set of realistic full protein redesign problems. We observe that it is possible to generate provably diverse libraries in reasonable time and that the produced libraries do enhance the Native Sequence Recovery, a traditional measure of design methods reliability.

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 generalised arc consistency algorithm for the acyclicity constraint. We embed these in the constraint programmingbased branch-and-bound solver CPBayes and show that, despite being suboptimal, they improve performance by orders of magnitude. The resulting solver also compares favourably 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.

By concisely representing a joint function of many variables as the combination of small functions, discrete graphical models (GMs) provide a powerful framework to analyze stochastic and deterministic systems of interacting variables. One of the main queries on such models is to identify the extremum of this joint function. This is known as the Weighted Constraint Satisfaction Problem (WCSP) on deterministic Cost Function Networks and as Maximum a Posteriori (MAP) inference on stochastic Markov Random Fields. Algorithms for approximate WCSP inference typically rely on local consistency algorithms or belief propagation. These methods are intimately related to linear programming (LP) relaxations and often coupled with reparametrizations defined by the dual solution of the associated LP. Since the seminal work of Goemans and Williamson, it is well understood that convex SDP relaxations can provide superior guarantees to LP. But the inherent computational cost of interior point methods has limited their application. The situation has improved with the introduction of non-convex Burer-Monteiro style methods which are well suited to handle the SDP relaxation of combinatorial problems with binary variables (such as MAXCUT, MaxSAT or MAP/Ising). We compute low rank SDP upper and lower bounds for discrete pairwise graphical models with arbitrary number of values and arbitrary binary cost functions by extending a Burer-Monteiro style method based on row-by-row updates. We consider a traditional dualized constraint approach and a dedicated Block Coordinate Descent approach which avoids introducing large penalty coefficients to the formulation. On increasingly hard and dense WCSP/CFN instances, we observe that the BCD approach can outperform the dualized approach and provide tighter bounds than local consistencies/convergent message passing approaches.