Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms

Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization

A Microstructure-based Family of Tractable Classes for CSPs

International audienceThe study of tractable classes is an important issue in Artificial Intelligence, especially in Constraint Satisfaction Problems. In this context, the Broken Triangle Property (BTP) is a state-of-the-art microstructure-based tractable class which generalizes well-known and previously-defined tractable classes, notably the set of instances whose constraint graph is a tree. In this paper, we propose to extend and to generalize this class using a more general approach based on a parameter k which is a given constant. To this end, we introduce the k-BTP property (and the class of instances satisfying this property) such that we have 2-BTP = BTP, and for k > 2, k-BTP is a relaxation of BTP in the sense that k-BTP is a subset of (k + 1)-BTP. Moreover, we show that if k-TW is the class of instances having tree-width bounded by a constant k, then k-TW is a subset of (k + 1)-BTP. Concerning tractability, we show that instances satisfying k-BTP and which are strong k-consistent are tractable, that is, can be recognized and solved in polynomial time. We also study the relationship between k-BTP and the approach of Naanaa who proposed a set-theoretical tool, known as the directional rank, to extend tractable classes in a parameterized way. Finally we propose an experimental study of 3-BTP which shows the practical interest of this class, particularly w.r.t. the practical solving of instances satisfying 3-BTP and for other instances, w.r.t. to backdoors based on this tractable class

On the Efficiency of Backtracking Algorithms for Binary Constraint Satisfaction Problems

International audienceThe question of tractable classes of constraint satisfaction problems (CSPs) has been studied for a long time, and is now a very active research domain. However, studies of tractable classes are typically very theoretical. They usually introduce classes of instances together with polynomial time algorithms for recognizing and solving them, and the algorithms can be used only for the new class. In this paper, we address the issue of tractable classes of CSPs from a different perspective. We investigate the complexity of classical, generic algorithms for solving CSPs (such as Forward Checking). We introduce a new parameter for measuring their complexity and derive new complexity bounds. By relating the complexity of CSP algorithms to graph-theoretic parameters, our analysis allows us to point at new tractable classes, which can be solved directly by the usual CSP algorithms in polynomial time, and without the need to recognize the classes in advance

Reconfiguration in bounded bandwidth and treedepth

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.Comment: 14 page

Size-Change Termination, Monotonicity Constraints and Ranking Functions

Size-Change Termination (SCT) is a method of proving program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x>y' and x>=y' are allowed, we have size-change graphs. Both theory and practice are now more evolved in this restricted framework then in the general framework of monotonicity constraints. This paper shows that it is possible to extend and adapt some theory from the domain of size-change graphs to the general case, thus complementing previous work on monotonicity constraints. In particular, we present precise decision procedures for termination; and we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singly-exponential time, which is better than what has been published so far even for size-change graphs.Comment: revised version of September 2