Valued Constraint Satisfaction Problems over Infinite Domains

The object of the thesis is the computational complexity of certain combinatorial optimisation problems called \emph{valued constraint satisfaction problems}, or \emph{VCSPs} for short. The requirements and optimisation criteria of these problems are expressed by sums of \emph{(valued) constraints} (also called \emph{cost functions}). More precisely, the input of a VCSP consists of a finite set of variables, a finite set of cost functions that depend on these variables, and a cost $u$; the task is to find values for the variables such that the sum of the cost functions is at most $u$. By restricting the set of possible cost functions in the input, a great variety of computational optimisation problems can be modelled as VCSPs. Recently, the computational complexity of all VCSPs for finite sets of cost functions over a finite domain has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear (PL) and piecewise linear homogeneous (PLH) cost functions. The VCSP for a finite set of PLH cost functions can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by (polynomial-time many-one) reducing the problem to a finite-domain VCSP which can be solved using a linear programming relaxation. We apply this result to show the polynomial-time tractability of VCSPs for {\it submodular} PLH cost functions, for {\it convex} PLH cost functions, and for {\it componentwise increasing} PLH cost functions; in fact, we show that submodular PLH functions and componentwise increasing PLH functions form maximally tractable classes of PLH cost functions. We define the notion of {\it expressive power} for sets of cost functions over arbitrary domains, and discuss the relation between the expressive power and the set of fractional operations improving the same set of cost functions over an arbitrary countable domain. Finally, we provide a polynomial-time algorithm solving the restriction of the VCSP for {\it all} PL cost functions to a fixed number of variables

On Spatial Conjunction as Second-Order Logic

Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its precise expressive power compared to traditional logical constructs was not previously known. In this paper we establish the expressive power of spatial conjunction. We construct an embedding from first-order logic with spatial conjunction into second-order logic, and more surprisingly, an embedding from full second order logic into first-order logic with spatial conjunction. These embeddings show that the satisfiability of formulas in first-order logic with spatial conjunction is equivalent to the satisfiability of formulas in second-order logic. These results explain the great expressive power of spatial conjunction and can be used to show that adding unrestricted spatial conjunction to a decidable logic leads to an undecidable logic. As one example, we show that adding unrestricted spatial conjunction to two-variable logic leads to undecidability. On the side of decidability, the embedding into second-order logic immediately implies the decidability of first-order logic with a form of spatial conjunction over trees. The embedding into spatial conjunction also has useful consequences: because a restricted form of spatial conjunction in two-variable logic preserves decidability, we obtain that a correspondingly restricted form of second-order quantification in two-variable logic is decidable. The resulting language generalizes the first-order theory of boolean algebra over sets and is useful in reasoning about the contents of data structures in object-oriented languages.Comment: 16 page

Half-integrality, LP-branching and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). However, although interesting results have been achieved, the methods require the underlying polytope to have very restrictive properties (half-integrality and persistence), which are known only for few problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such relaxations, we provide a much broader class of half-integral polytopes with the required properties, unifying and extending previously known cases. In addition to the insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access. All these significantly improve on previous results. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network flow-based approach to solve some cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA paper

On Algorithms and Complexity for Sets with Cardinality Constraints

Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate properties: relationships between the typestates of objects can be expressed as subset and disjointness relations on sets, and elements of sets can be represented as sets of cardinality one. Motivated by these applications, this paper presents new algorithms and new complexity results for constraints on sets and their cardinalities. We study several classes of constraints and demonstrate a trade-off between their expressive power and their complexity. Our first result concerns a quantifier-free fragment of Boolean Algebra with Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for reducing the satisfiability of sets with symbolic cardinalities to constraints on constant cardinalities, and give a polynomial-space algorithm for the resulting problem. In a quest for more efficient fragments, we identify several subclasses of sets with cardinality constraints whose satisfiability is NP-hard. Finally, we identify a class of constraints that has polynomial-time satisfiability and entailment problems and can serve as a foundation for efficient program analysis.Comment: 20 pages. 12 figure

PDDL2.1: An extension of PDDL for expressing temporal planning domains

In recent years research in the planning community has moved increasingly towards application of planners to realistic problems involving both time and many types of resources. For example, interest in planning demonstrated by the space research community has inspired work in observation scheduling, planetary rover ex ploration and spacecraft control domains. Other temporal and resource-intensive domains including logistics planning, plant control and manufacturing have also helped to focus the community on the modelling and reasoning issues that must be confronted to make planning technology meet the challenges of application. The International Planning Competitions have acted as an important motivating force behind the progress that has been made in planning since 1998. The third competition (held in 2002) set the planning community the challenge of handling time and numeric resources. This necessitated the development of a modelling language capable of expressing temporal and numeric properties of planning domains. In this paper we describe the language, PDDL2.1, that was used in the competition. We describe the syntax of the language, its formal semantics and the validation of concurrent plans. We observe that PDDL2.1 has considerable modelling power --- exceeding the capabilities of current planning technology --- and presents a number of important challenges to the research community

Progress in AI Planning Research and Applications

Planning has made significant progress since its inception in the 1970s, in terms both of the efficiency and sophistication of its algorithms and representations and its potential for application to real problems. In this paper we sketch the foundations of planning as a sub-field of Artificial Intelligence and the history of its development over the past three decades. Then some of the recent achievements within the field are discussed and provided some experimental data demonstrating the progress that has been made in the application of general planners to realistic and complex problems. The paper concludes by identifying some of the open issues that remain as important challenges for future research in planning

Heuristic Ranking in Tightly Coupled Probabilistic Description Logics

The Semantic Web effort has steadily been gaining traction in the recent years. In particular,Web search companies are recently realizing that their products need to evolve towards having richer semantic search capabilities. Description logics (DLs) have been adopted as the formal underpinnings for Semantic Web languages used in describing ontologies. Reasoning under uncertainty has recently taken a leading role in this arena, given the nature of data found on theWeb. In this paper, we present a probabilistic extension of the DL EL++ (which underlies the OWL2 EL profile) using Markov logic networks (MLNs) as probabilistic semantics. This extension is tightly coupled, meaning that probabilistic annotations in formulas can refer to objects in the ontology. We show that, even though the tightly coupled nature of our language means that many basic operations are data-intractable, we can leverage a sublanguage of MLNs that allows to rank the atomic consequences of an ontology relative to their probability values (called ranking queries) even when these values are not fully computed. We present an anytime algorithm to answer ranking queries, and provide an upper bound on the error that it incurs, as well as a criterion to decide when results are guaranteed to be correct.Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012

LTLf/LDLf Non-Markovian Rewards

In Markov Decision Processes (MDPs), the reward obtained in a state is Markovian, i.e., depends on the last state and action. This dependency makes it difficult to reward more interesting long-term behaviors, such as always closing a door after it has been opened, or providing coffee only following a request. Extending MDPs to handle non-Markovian reward functions was the subject of two previous lines of work. Both use LTL variants to specify the reward function and then compile the new model back into a Markovian model. Building on recent progress in temporal logics over finite traces, we adopt LDLf for specifying non-Markovian rewards and provide an elegant automata construction for building a Markovian model, which extends that of previous work and offers strong minimality and compositionality guarantees