scholarly journals Semantic Width and the Fixed-Parameter Tractability of Constraint Satisfaction Problems

2020 ◽  
Author(s):  
Hubie Chen ◽  
Georg Gottlob ◽  
Matthias Lanzinger ◽  
Reinhard Pichler
Keyword(s):  
Constraint Satisfaction ◽  
Fundamental Problem ◽  
Constraint Satisfaction Problems ◽  
Fixed Parameter Tractability ◽  
Scheduling Problems ◽  
Fixed Parameter Tractable ◽  
Database Theory ◽  
Formal Framework ◽  
Fixed Parameter

Constraint satisfaction problems (CSPs) are an important formal framework for the uniform treatment of various prominent AI tasks, e.g., coloring or scheduling problems. Solving CSPs is, in general, known to be NP-complete and fixed-parameter intractable when parameterized by their constraint scopes. We give a characterization of those classes of CSPs for which the problem becomes fixed-parameter tractable. Our characterization significantly increases the utility of the CSP framework by making it possible to decide the fixed-parameter tractability of problems via their CSP formulations. We further extend our characterization to the evaluation of unions of conjunctive queries, a fundamental problem in databases. Furthermore, we provide some new insight on the frontier of PTIME solvability of CSPs. In particular, we observe that bounded fractional hypertree width is more general than bounded hypertree width only for classes that exhibit a certain type of exponential growth. The presented work resolves a long-standing open problem and yields powerful new tools for complexity research in AI and database theory.

scholarly journals Constraint programming viewed as rule-based programming

2001 ◽  
Vol 1 (6) ◽  
pp. 713-750 ◽  
Author(s):  
KRZYSZTOF R. APT ◽  
ERIC MONFROY
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Programming ◽  
Constraint Satisfaction Problems ◽  
Rule Based ◽  
Local Consistency ◽  
Arc Consistency ◽  
Computation Process ◽  
Rule Based Approach ◽  
Rule Consistency

We study here a natural situation when constraint programming can be entirely reduced to rule-based programming. To this end we explain first how one can compute on constraint satisfaction problems using rules represented by simple first-order formulas. Then we consider constraint satisfaction problems that are based on predefined, explicitly given constraints. To solve them we first derive rules from these explicitly given constraints and limit the computation process to a repeated application of these rules, combined with labeling. We consider two types of rule here. The first type, that we call equality rules, leads to a new notion of local consistency, called rule consistency that turns out to be weaker than arc consistency for constraints of arbitrary arity (called hyper-arc consistency in Marriott & Stuckey (1998)). For Boolean constraints rule consistency coincides with the closure under the well-known propagation rules for Boolean constraints. The second type of rules, that we call membership rules, yields a rule-based characterization of arc consistency. To show feasibility of this rule-based approach to constraint programming, we show how both types of rules can be automatically generated, as CHR rules of Frühwirth (1995). This yields an implementation of this approach to programming by means of constraint logic programming. We illustrate the usefulness of this approach to constraint programming by discussing various examples, including Boolean constraints, two typical examples of many valued logics, constraints dealing with Waltz's language for describing polyhedral scenes, and Allen's qualitative approach to temporal logic.

scholarly journals Threshold Treewidth and Hypertree Width

2020 ◽  
Author(s):  
Robert Ganian ◽  
Andre Schidler ◽  
Manuel Sorge ◽  
Stefan Szeider
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Structural Parameters ◽  
Exact Methods ◽  
Fixed Parameter Tractable ◽  
Computational Costs ◽  
Fixed Parameter ◽  
Account Information ◽  
The Given ◽  
Theoretical Results

Treewidth and hypertree width have proven to be highly successful structural parameters in the context of the Constraint Satisfaction Problem (CSP). When either of these parameters is bounded by a constant, then CSP becomes solvable in polynomial time. However, here the order of the polynomial in the running time depends on the width, and this is known to be unavoidable; therefore, the problem is not fixed-parameter tractable parameterized by either of these width measures. Here we introduce an enhancement of tree and hypertree width through a novel notion of thresholds, allowing the associated decompositions to take into account information about the computational costs associated with solving the given CSP instance. Aside from introducing these notions, we obtain efficient theoretical as well as empirical algorithms for computing threshold treewidth and hypertree width and show that these parameters give rise to fixed-parameter algorithms for CSP as well as other, more general problems. We complement our theoretical results with experimental evaluations in terms of heuristics as well as exact methods based on SAT/SMT encodings.

On finite Taylor algebras

2016 ◽  
Vol 26 (08) ◽  
pp. 1547-1571
Author(s):  
Alexander Wires
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problems ◽  
Algebraic Constructions ◽  
Bounded Width

We establish a new hereditary characterization of finite idempotent Taylor algebras. This generalizes the algebraic constructions which figured in the recent successful characterization of the correctness of the bounded width algorithm for constraint satisfaction problems by finite idempotent algebras generating congruence meet-semidistributive varieties. We introduce the cyclic reduct of a finite Taylor algebra and prove a collapse of Mal’cev conditions.

scholarly journals On the complexity of Gödel's proof predicate

2010 ◽  
Vol 75 (1) ◽  
pp. 239-254 ◽  
Author(s):  
Yijia Chen ◽  
Jörg Flum
Keyword(s):  
Classical Problem ◽  
Order Logic ◽  
First Order Logic ◽  
Fixed Parameter Tractability ◽  
Fixed Parameter Tractable ◽  
First Order ◽  
Fixed Parameter ◽  
Parameterized Problem ◽  
Computable Bound ◽  
Increasing Functions

AbstractThe undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(hO(1)) ≠ NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

scholarly journals Replaceability for Constraint Satisfaction Problems: Algorithms, Inference, and Complexity Patterns

10.29007/wrp9 ◽  
2018 ◽  
Author(s):  
Richard Wallace
Keyword(s):  
Constraint Satisfaction ◽  
Relative Effectiveness ◽  
Constraint Satisfaction Problems ◽  
Scheduling Problems ◽  
Problem Space ◽  
All Solutions ◽  
Disjunctive Constraints ◽  
Neighbourhood Search ◽  
Structured Problems ◽  
Full Search

Replaceability is a form of generalized substitutability whose features make it potentially of great importance for problem simplification. It differs from simple substitutability in that it only requires that substitutable values exist for every solution containing a given value without requiring that the former always be the same. This is the most general form of substitutability that allows inferences from local to global versions of this property. Building on earlier work, this study first establishes that algorithms for localized replaceability (consistent neighbourhood replaceability or CNR algorithms) based on all-solutions neighbourhood search outperform other replaceability algorithms by several orders of magnitude. It also examines the relative effectiveness of different forms of depth-first CNR algorithms. Secondly, it demonstrates an apparent complexity ridge, which does not occur at the same place in the problem space as the complexity areas for consistency or full search algorithms. Thirdly, it continues the study of methods for inferring replaceability in structured problems in order to improve efficiency. Here, it is shown that some strategies for inferring replaceable values can be extended to disjunctive constraints in scheduling problems.

scholarly journals The 2008 Classic Paper Award: Summary and Significance

AI Magazine ◽  
2010 ◽  
Vol 31 (4) ◽  
pp. 109
Author(s):  
Peter Friedland
Keyword(s):  
Local Search ◽  
Constraint Satisfaction ◽  
Wide Class ◽  
Large Scale ◽  
Constraint Satisfaction Problems ◽  
Paper Award ◽  
Scheduling Problems ◽  
Repair Method ◽  
Mark Johnston ◽  
Classic Paper

We at the NASA laboratory believed that our best work came when we simultaneously advanced AI theory and provided immediately usable solutions for current NASA problems. “Solving Large-Scale Constraint Satisfaction and Scheduling Problems Using a Heuristic Repair Method,” by Steve Minton, Mark Johnston, Andy Phillips, and Phil Laird clearly achieved both. It proved that local search and repair was applicable to a wide class of constraint satisfaction problems and clearly explicated the theory behind that proof.

scholarly journals The Parameterized Complexity of Connected Fair Division

2021 ◽  
Author(s):  
Argyrios Deligkas ◽  
Eduard Eiben ◽  
Robert Ganian ◽  
Thekla Hamm ◽  
Sebastian Ordyniak
Keyword(s):  
Lower Bounds ◽  
Parameterized Complexity ◽  
Fundamental Problem ◽  
Fair Division ◽  
Division Problem ◽  
Additional Parameter ◽  
Fixed Parameter Tractability ◽  
Connected Subgraph ◽  
Fixed Parameter ◽  
New Algorithms

We study the Connected Fair Division problem (CFD), which generalizes the fundamental problem of fairly allocating resources to agents by requiring that the items allocated to each agent form a connected subgraph in a provided item graph G. We expand on previous results by providing a comprehensive complexity-theoretic understanding of CFD based on several new algorithms and lower bounds while taking into account several well-established notions of fairness: proportionality, envy-freeness, EF1 and EFX. In particular, we show that to achieve tractability, one needs to restrict both the agents and the item graph in a meaningful way. We design (XP)-algorithms for the problem parameterized by (1) clique-width of G plus the number of agents and (2) treewidth of G plus the number of agent types, along with corresponding lower bounds. Finally, we show that to achieve fixed-parameter tractability, one needs to not only use a more restrictive parameterization of G, but also include the maximum item valuation as an additional parameter.

scholarly journals Truth-Preservation under Fuzzy pp-Formulas

2019 ◽  
Vol 27 (Supp01) ◽  
pp. 89-105
Author(s):  
Pilar Dellunde ◽  
Amanda Vidal
Keyword(s):  
Computer Science ◽  
Constraint Satisfaction ◽  
Classical Logic ◽  
Constraint Satisfaction Problems ◽  
Algebraic Characterization ◽  
Theoretic Approach ◽  
Direct Products ◽  
Fuzzy Constraint ◽  
Fuzzy Constraint Satisfaction

How can non-classical logic contribute to the analysis of complexity in computer science? In this paper, we give a step towards this question, taking a logical model-theoretic approach to the analysis of complexity in fuzzy constraint satisfaction. We study fuzzy positive-primitive sentences, and we present an algebraic characterization of classes axiomatized by this kind of sentences in terms of homomorphisms and direct products. The ultimate goal is to study the expressiveness and reasoning mechanisms of non-classical languages, with respect to constraint satisfaction problems and, in general, in modelling decision scenarios.

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