Multi-objective evolutionary search strategies in constraint programming

Constraint programming offers a declarative approach to solving problems modeled as constraint satisfaction problems (CSPs). However, the precise specification of a set of constraints is sometimes not available, but may have to be learned, for instance, from a set of examples of its solutions and non-solutions. In general, one may wish to learn generalized CSPs involving classical, fuzzy, weighted or probabilistic constraints, for example. This paper introduces a unifying framework for CSP learning. The framework is generic in that it can be instantiated to obtain specific formulations for learning classical, fuzzy, weighted or probabilistic CSPs. In particular, a new formulation for classical CSP learning, which minimizes the number of examples violated by candidate CSPs, is obtained by instantiating the framework. This formulation is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable using many optimization tools.

In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras.

Practical implementation of random binning is one of the key challenges in achieving the largest available rate regions for many multiuser channels. This paper explores the use of low-density parity-check (LDPC) like codes for a particular kind of deterministic broadcast channel called the Blackwell channel and illustrates that random linear codes can be used to construct practical binning schemes at rates close to the capacity region of the Blackwell channel. The key ingredient is an encoding algorithm known as which is a generalization of the well-known belief propagation algorithm for LDPC codes. Survey propagation has been previously devised for a class of constraint satisfaction problems called K-SAT. This paper shows that the encoding problem for the Blackwell channel contains the same features as the constraint satisfaction problem and that the survey propagation algorithm, when concatenated with an outer error correcting code, works well at rates close to the Blackwell channel capacity region

Constraint Satisfaction with a Multi-Dimensional Domain

The expressiveness of First Order Logic (FOL) languages is severely counterbalanced by the complexity of matching formulas on a universe. Matching is an instance of the class of Constraint Satisfaction Problems (CSP), which have shown to undergo a phase transition with respect to two order parameters: constraint density and constraint tightness. This paper analyzes the problem of satisfying FOL Horn clauses in the light of these recent results. By means of an extensive experimental analysis, we show how Horn clause verification exhibits a typical phase transition with respect to the number of binary (or greater arity) predicates, and with respect to the ratio between the number of constants in the universe and the cardinality of the basic predicates extension.

We introduce a generalization of the spreadsheet paradigm, called Knowledgesheet, for solving a class of constraint satisfaction problems. The traditional spreadsheet paradigm is based on attaching arithmetic expressions to individual cells and then evaluating them; our Knowledgesheet interface instead allows finite domain constraints to be attached to individual cells that are then solved to obtain a solution. This extension provides an easy-to-use interface for solving a large class of constraint satisfaction problems--those whose specification and solution conforms to a 2-dimensional structure, e.g., scheduling problems, timetabling problems, etc. A prototype for the Knowledgesheet has been developed and applied to solve many different types of problems.

The goal of this paper is to develop models and methods that use complementary strengths of Mixed Integer Linear Programming (MILP) and Constraint Programming (CP) techniques to solve problems that are otherwise intractable if solved using either of the two methods. The class of problems considered in this paper have the characteristic that only a subset of the binary variables have non-zero objective function coefficients if modeled as an MILP. This class of problems is formulated as a hybrid MILP/CP model that involves some of the MILP constraints, a reduced set of the CP constraints, and equivalence relations between the MILP and the CP variables. An MILP/CP based decomposition method and an LP/CP-based branch-and-bound algorithm are proposed to solve these hybrid models. Both these algorithms rely on the same relaxed MILP and feasibility CP problems. An application example is considered in which the least-cost schedule has to be derived for processing a set of orders with release and due dates using a set of dissimilar parallel machines. It is shown that this problem can be modeled as an MILP, a CP, a combined MILP-CP OPL model (Van Hentenryck 1999), and a hybrid MILP/CP model. The computational performance of these models for several sets shows that the hybrid MILP/CP model can achieve two to three orders of magnitude reduction in CPU time.

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called “islands of tractability.” A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language , whereas a constraint language is conservative if it contains all unary relations. Schaefer’s famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable constraint languages over a Boolean domain of values. Since then, many extensions and generalizations of this result have been obtained. Recently, Bulatov (TOCL 2011, JACM 2013) gave a full characterization of all islands of tractability for CSP and the counting version #CSP that are defined in terms of conservative constraint languages. This article addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages is not tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose, we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated, moves the instance to an island of tractability, that is, to a tractable class of instances. We consider strong backdoors into scattered classes , consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipelago of tractability . The main difficulty lies in finding a strong backdoor of given size k ; once it is found, we can try all possible instantiations of the backdoor variables and apply the polynomial time algorithms associated with the islands of tractability on the list component-wise. Our main result is an algorithm that, given a CSP instance with n variables, finds in time f ( k ) n O (1) a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size k or correctly decides that there is not such a backdoor. This also gives the running time for solving (#)CSP, provided that (#)CSP is polynomial-time tractable for the considered constraint languages. Our result makes significant progress towards the main goal of the backdoor-based approach to CSPs—the identification of maximal base classes for which small backdoors can be detected efficiently.

A note on some collapse results of valued constraints

The aim of the work is to create a scenario modeling technique based on adaptation of metaheuristic multi-objective optimization algorithms to construct Pareto-optimal investment portfolios taking into account multiple restrictions on the company's portfolio. Investment portfolio is a tool for company's resources management. The process of portfolio construction through a combination of individual investment projects aims to optimize available resources utilization. Currently, there is no single approach to determining the optimal portfolio, since portfolio requirements depend on a particular company. Due to its simplicity in terms of implementation, approaches based on the ranking of projects according to individual technical and economic indicators are widely used. However, if a company needs to take into account several indicators, then this approach is not applicable. An effective solution to this problem is to create a technique that allows to find Pareto-optimal portfolios. In this work, the problem statement and analysis of existing solving approaches were carried out. Based on that, methodology of scenario modeling of a company's portfolio was created. The posed problem belongs to the class of NP-hard problems and for its efficient solving it is necessary to use metaheuristic algorithms. These algorithms include genetic algorithms, variable neighborhood search, bees algorithm, etc. Genetic algorithm NSGA-II is widely used for solving multi-objective optimization problems due to its high performance and ease of use. This algorithm was adapted to solve proposed problem. Based on it, a methodology was developed for the formation of Pareto-optimal investment portfolios, taking into account financial and infrastructural restrictions. Testing of the methodology was carried out on a dataset of synthetic projects simulating real investment projects. Various combinations of this projects simulated investment portfolios. The optimization objectives were maximization of net present value (NPV), maximization of cumulative oil production and minimization of total investments. In order to control the quality of the results obtained using genetic algorithm, the problem was also solved by using variable neighborhood search. However, the quality of the results obtained using genetic algorithm was higher. Analysis of the results allows us to consider the use of multi-objective optimization as a tool for forming a pool of investment portfolios, which provides flexible company management. In the proposed work, it is shown that the problem of scenario-based modeling of company's investment portfolio can be considered as a multi-objective optimization problem. Based on the existing metaheuristic algorithms, a Pareto-optimal investment portfolio search methodology has been developed. Proposed methodology can effectively solve the problem of managing company resources.

A novel multi-objective particle swarm optimization with multiple search strategies

In the frame of classical constraint satisfaction problems (CSPs), the backtrack tree search, combined with learning methods, presents a double advantage: for static solving, it improves the search speed by avoiding redundant explorations; for dynamic solving (after a slight change of the problem) it reuses the previous searches to build a new solution quickly. Backtrack reasoning concludes the rejection of certain combinatorial choices. Nogood Recording memorizes these choices in order to not reproduce. We aim to use Nogood Recording in the wider scope of the CSP framework (VCSP) to enhance the branch and bound algorithm. Therefore, nogoods are used to increase the lower bound used by the branch and bound to prune the search. This issue leads to the definition of the Valued Nogoods and their use. This study focuses particularly on penalty and dynamic VCSPs which require special developments. However our results give an extension of the Nogood Recording to the general VCSP framework.

The modal satisfiability problem is solved either by using a specifically designed algorithm, or by translating the modal logic formula into an instance of a different class of problem, such as a first-order logic problem, a propositional satisfiability problem, or, more recently, a constraint satisfaction problem. In the latter approach, the modal formula is translated into layered propositional formulae. Each layer is translated into a constraint satisfaction problem which is solved using a constraint solver. We extend this approach to the modal logics KT and S4 and introduce a range of optimizations of the basic prototype. The results compare favorably with those of other solvers, and support the adoption of constraint programming as implementation platform for modal and other related satisfiability solvers.

: The primary objective of this project has been the development of systems that reason in dynamic and open-ended environment and that use networks as their primary representation language. The focus of our research has been temporal reasoning, neural networks, truth maintenance, and default reasoning. This investigation has led to several basic results: the expressiveness of constraint networks was analyzed, tractable classes of constraint satisfaction problems were identified and effective processing techniques were developed.

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on theform of the constraints can ensure tractability. The standard way to parameterize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The mainproblem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a core of a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete. In the paper we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.