Backdoor Sets for CSP

A backdoor set of a CSP instance is a set of variables whose instantiation moves the instance into a fixed class of tractable instances (an island of tractability). An interesting algorithmic task is to find a small backdoor set efficiently: once it is found we can solve the instance by solving a number of tractable instances. Parameterized complexity provides an adequate framework for studying and solving this algorithmic task, where the size of the backdoor set provides a natural parameter. In this survey we present some recent parameterized complexity results on CSP backdoor sets, focusing on backdoor sets into islands of tractability that are defined in terms of constraint languages

Structural Decompositions for Problems with Global Constraints

A wide range of problems can be modelled as constraint satisfaction problems (CSPs), that is, a set of constraints that must be satisfied simultaneously. Constraints can either be represented extensionally, by explicitly listing allowed combinations of values, or implicitly, by special-purpose algorithms provided by a solver. Such implicitly represented constraints, known as global constraints, are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. In recent years, a variety of restrictions on the structure of CSP instances have been shown to yield tractable classes of CSPs. However, most such restrictions fail to guarantee tractability for CSPs with global constraints. We therefore study the applicability of structural restrictions to instances with such constraints. We show that when the number of solutions to a CSP instance is bounded in key parts of the problem, structural restrictions can be used to derive new tractable classes. Furthermore, we show that this result extends to combinations of instances drawn from known tractable classes, as well as to CSP instances where constraints assign costs to satisfying assignments.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s10601-015-9181-

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP

Backdoors to planning

Backdoors measure the distance to tractable fragments and have become an important tool to find fixed-parameter tractable (fpt) algorithms for hard problems in AI and beyond. Despite their success, backdoors have not been used for planning, a central problem in AI that has a high computational complexity. In this work, we introduce two notions of backdoors building upon the causal graph. We analyze the complexity of finding a small backdoor (detection) and using the backdoor to solve the problem (evaluation) in the light of planning with (un)bounded plan length/domain of the variables. For each setting we present either an fpt-result or rule out the existence thereof by showing parameterized intractability. For several interesting cases we achieve the most desirable outcome: detection and evaluation are fpt. In addition, we explore the power of polynomial preprocessing for all fpt-results, i.e., we investigate whether polynomial kernels exist. We show that for the detection problems, polynomial kernels exist whereas we rule out the existence of polynomial kernels for the evaluation problems

Harnessing tractability in constraint satisfaction problems

The Constraint Satisfaction Problem (CSP) is a fundamental NP-complete problem with many applications in artificial intelligence. This problem has enjoyed considerable scientific attention in the past decades due to its practical usefulness and the deep theoretical questions it relates to. However, there is a wide gap between practitioners, who develop solving techniques that are efficient for industrial instances but exponential in the worst case, and theorists who design sophisticated polynomial-time algorithms for restrictions of CSP defined by certain algebraic properties. In this thesis we attempt to bridge this gap by providing polynomial-time algorithms to test for membership in a selection of major tractable classes. Even if the instance does not belong to one of these classes, we investigate the possibility of decomposing efficiently a CSP instance into tractable subproblems through the lens of parameterized complexity. Finally, we propose a general framework to adapt the concept of kernelization, central to parameterized complexity but hitherto rarely used in practice, to the context of constraint reasoning. Preliminary experiments on this last contribution show promising results

On Formal Methods for Large-Scale Product Configuration

<p>In product development companies mass customization is widely used to achieve better customer satisfaction while keeping costs down. To efficiently implement mass customization, product platforms are often used. A product platform allows building a wide range of products from a set of predefined components. The process of matching these components to customers' needs is called product configuration. Not all components can be combined with each other due to restrictions of various kinds, for example, geometrical, marketing and legal reasons. Product design engineers develop configuration constraints to describe such restrictions. The number of constraints and the complexity of the relations between them are immense for complex product like a vehicle. Thus, it is both error-prone and time consuming to analyze, author and verify the constraints manually. Software tools based on formal methods can help engineers to avoid making errors when working with configuration constraints, thus design a correct product faster.</p> <p>This thesis introduces a number of formal methods to help engineers maintain, verify and analyze product configuration constraints. These methods provide automatic verification of constraints and computational support for analyzing and refactoring constraints. The methods also allow verifying the correctness of one specific type of constraints, item usage rules, for sets of mutually-exclusive required items, and automatic verification of equivalence of different formulations of the constraints. The thesis also introduces three methods for efficient enumeration of valid partial configurations, with benchmarking of the methods on an industrial dataset.</p> <p>Handling large-scale industrial product configuration problems demands high efficiency from the software methods. This thesis investigates a number of search-based and knowledge-compilation-based methods for working with large product configuration instances, including Boolean satisfiability solvers, binary decision diagrams and decomposable negation normal form. This thesis also proposes a novel method based on supervisory control theory for efficient reasoning about product configuration data. The methods were implemented in a tool, to investigate the applicability of the methods for handling large product configuration problems. It was found that search-based Boolean satisfiability solvers with incremental capabilities are well suited for industrial configuration problems.</p> <p>The methods proposed in this thesis exhibit good performance on practical configuration problems, and have a potential to be implemented in industry to support product design engineers in creating and maintaining configuration constraints, and speed up the development of product platforms and new products.</p

Exploiting Structure In Combinatorial Problems With Applications In Computational Sustainability

Combinatorial decision and optimization problems are at the core of many tasks with practical importance in areas as diverse as planning and scheduling, supply chain management, hardware and software verification, electronic commerce, and computational biology. Another important source of combinatorial problems is the newly emerging field of computational sustainability, which addresses decision-making aimed at balancing social, economic and environmental needs to guarantee the long-term prosperity of life on our planet. This dissertation studies different forms of problem structure that can be exploited in developing scalable algorithmic techniques capable of addressing large real-world combinatorial problems. There are three major contributions in this work: 1) We study a form of hidden problem structure called a backdoor, a set of key decision variables that captures the combinatorics of the problem, and reveal that many real-world problems encoded as Boolean satisfiability or mixed-integer linear programs contain small backdoors. We study backdoors both theoretically and empirically and characterize important tradeoffs between the computational complexity of finding backdoors and their effectiveness in capturing problem structure succinctly. 2) We contribute several domain-specific mathematical formulations and algorithmic techniques that exploit specific aspects of problem structure arising in budget-constrained conservation planning for wildlife habitat connectivity. Our solution approaches scale to real-world conservation settings and provide important decision-support tools for cost-benefit analysis. 3) We propose a new survey-planning methodology to assist in the construction of accurate predictive models, which are especially relevant in sustainability areas such as species- distribution prediction and climate-change impact studies. In particular, we design a technique that takes advantage of submodularity, a structural property of the function to be optimized, and results in a polynomial-time procedure with approximation guarantees

Parametrised enumeration

In this thesis, we develop a framework of parametrised enumeration complexity. At first, we provide the reader with preliminary notions such as machine models and complexity classes besides proving them to be well-chosen. Then, we study the interplay and the landscape of these classes and present connections to classical enumeration classes. Afterwards, we translate the fundamental methods of kernelisation and self-reducibility into equivalent techniques in the setting of parametrised enumeration. Subsequently, we illustrate the introduced classes by investigating the parametrised enumeration complexity of Max-Ones-SAT and strong backdoor sets as well as sharpen the first result by presenting a dichotomy theorem for Max-Ones-SAT. After this, we extend the definitions of parametrised enumeration algorithms by allowing orders on the solution space. In this context, we study the relations ``order by size'' and ``lexicographic order'' for graph modification problems and observe a trade-off between enumeration delay and space requirements of enumeration algorithms. These results then yield an enumeration technique for generalised modification problems that is illustrated by applying this method to the problems closest string, weak and strong backdoor sets, and weighted satisfiability. Eventually, we consider the enumeration of satisfying teams of formulas of poor man's propositional dependence logic. There, we present an enumeration algorithm with FPT delay and exponential space which is one of the first enumeration complexity results of a problem in a team logic. Finally, we show how this algorithm can be modified such that only polynomial space is required, however, by increasing the delay to incremental FPT time.In diesem Werk begründen wir die Theorie der parametrisierten Enumeration, präsentieren die grundlegenden Definitionen und prüfen ihre Sinnhaftigkeit. Im nächsten Schritt, untersuchen wir das Zusammenspiel der eingeführten Komplexitätsklassen und zeigen Verbindungen zur klassischen Enumerationskomplexität auf. Anschließend übertragen wir die zwei fundamentalen Techniken der Kernelisierung und Selbstreduzierbarkeit in Entsprechungen in dem Gebiet der parametrisierten Enumeration. Schließlich untersuchen wir das Problem Max-Ones-SAT und das Problem der Aufzählung starker Backdoor-Mengen als typische Probleme in diesen Klassen. Die vorherigen Resultate zu Max-Ones-SAT werden anschließend in einem Dichotomie-Satz vervollständigt. Im nächsten Abschnitt erweitern wir die neuen Definitionen auf Ordnungen (auf dem Lösungsraum) und erforschen insbesondere die zwei Relationen \glqq Größenordnung\grqq\ und \glqq lexikographische Reihenfolge\grqq\ im Kontext von Graphen-Modifikationsproblemen. Hierbei scheint es, als müsste man zwischen Delay und Speicheranforderungen von Aufzählungsalgorithmen abwägen, wobei dies jedoch nicht abschließend gelöst werden kann. Aus den vorherigen Überlegungen wird schließlich ein generisches Enumerationsverfahren für allgemeine Modifikationsprobleme entwickelt und anhand der Probleme Closest String, schwacher und starker Backdoor-Mengen sowie gewichteter Erfüllbarkeit veranschaulicht. Im letzten Abschnitt betrachten wir die parametrisierte Enumerationskomplexität von Erfüllbarkeitsproblemen im Bereich der Poor Man's Propositional Dependence Logic und stellen einen Aufzählungsalgorithmus mit FPT Delay vor, der mit exponentiellem Platz arbeitet. Dies ist einer der ersten Aufzählungsalgorithmen im Bereich der Teamlogiken. Abschließend zeigen wir, wie dieser Algorithmus so modifiziert werden kann, dass nur polynomieller Speicherplatz benötigt wird, bezahlen jedoch diese Einsparung mit einem Anstieg des Delays auf inkrementelle FPT Zeit (IncFPT)