The power of Sherali-Adams relaxations for general-valued CSPs

We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems to optimality. The condition is that of bounded width which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many known valued constraint languages are tractable via the third level of the Sherali-Adams relaxation. For the known languages, this is a significantly simpler algorithm than the previously obtained ones. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Zivny [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301

Approximation for Maximum Surjective Constraint Satisfaction Problems

Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are computational problems where we are given a set of variables denoting values from a finite domain B and a set of constraints on the variables. A solution to such a problem is a surjective mapping from the set of variables to B such that the number of satisfied constraints is maximized. We study the approximation performance that can be acccchieved by algorithms for these problems, mainly by investigating their relation with Max-CSPs (which are the corresponding problems without the surjectivity requirement). Our work gives a complexity dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the assumption that there is a complexity dichotomy for Max-CSP(B) between PO and APX-complete, which has already been proved on the Boolean domain and 3-element domains

Developing a labelled object-relational constraint database architecture for the projection operator

Current relational databases have been developed in order to improve the handling of stored data, however, there are some types of information that have to be analysed for which no suitable tools are available. These new types of data can be represented and treated as constraints, allowing a set of data to be represented through equations, inequations and Boolean combinations of both. To this end, constraint databases were defined and some prototypes were developed. Since there are aspects that can be improved, we propose a new architecture called labelled object-relational constraint database (LORCDB). This provides more expressiveness, since the database is adapted in order to support more types of data, instead of the data having to be adapted to the database. In this paper, the projection operator of SQL is extended so that it works with linear and polynomial constraints and variables of constraints. In order to optimize query evaluation efficiency, some strategies and algorithms have been used to obtain an efficient query plan. Most work on constraint databases uses spatiotemporal data as case studies. However, this paper proposes model-based diagnosis since it is a highly potential research area, and model-based diagnosis permits more complicated queries than spatiotemporal examples. Our architecture permits the queries over constraints to be defined over different sets of variables by using symbolic substitution and elimination of variables.Ministerio de Ciencia y Tecnología DPI2006-15476-C02-0

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