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.

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 Novel Structural Parameters for Acyclic Planning Using Tree Embeddings

2018 ◽  
Author(s):  
Christer Bäckström ◽  
Peter Jonsson ◽  
Sebastian Ordyniak
Keyword(s):  
Computer Science ◽  
Bounded Domain ◽  
Parameterized Complexity ◽  
Structural Parameters ◽  
Domain Size ◽  
Fixed Parameter Tractable ◽  
Bounded Treewidth ◽  
Fixed Parameter ◽  
Causal Graphs ◽  
Computational Properties

We introduce two novel structural parameters for acyclic planning (planning restricted to instances with acyclic causal graphs): up-depth and down-depth. We show that cost-optimal acyclic planning restricted to instances with bounded domain size and bounded up- or down-depth can be solved in polynomial time. For example, many of the tractable subclasses based on polytrees are covered by our result. We analyze the parameterized complexity of planning with bounded up- and down-depth: in a certain sense, down-depth has better computational properties than up-depth. Finally, we show that computing up- and down-depth are fixed-parameter tractable problems, just as many other structural parameters that are used in computer science. We view our results as a natural step towards understanding the complexity of acyclic planning with bounded treewidth and other parameters.

scholarly journals On the Subexponential-Time Complexity of CSP

2015 ◽  
Vol 52 ◽  
pp. 203-234 ◽  
Author(s):  
Ronald De Haan ◽  
Iyad Kanj ◽  
Stefan Szeider
Keyword(s):  
Constraint Satisfaction ◽  
Time Complexity ◽  
Constraint Satisfaction Problem ◽  
Structural Parameters ◽  
Constraint Satisfaction Problems ◽  
Global Constraints ◽  
Brute Force ◽  
Subexponential Time ◽  
Complete Problems ◽  
Np Complete

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice. Acknowledging the NP-hardness of such problems, one has to look beyond polynomial time computation. The theory of subexponential-time complexity provides such a framework, and has been enjoying increasing popularity in complexity theory. An instance of the constraint satisfaction problem with n variables over a domain of d values can be solved by brute-force in dn steps (omitting a polynomial factor). In this paper we study the existence of subexponential-time algorithms, that is, algorithms running in do(n) steps, for various natural restrictions of the constraint satisfaction problem. We consider both the constraint satisfaction problem in which all the constraints are given extensionally as tables, and that in which all the constraints are given intensionally in the form of global constraints. We provide tight characterizations of the subexponential-time complexity of the aforementioned problems with respect to several natural structural parameters, which allows us to draw a detailed landscape of the subexponential-time complexity of the constraint satisfaction problem. Our analysis provides fundamental results indicating whether and when one can significantly improve on the brute-force search approach for solving the constraint satisfaction problem.

scholarly journals Iterative Plan Construction for the Workflow Satisfiability Problem

2014 ◽  
Vol 51 ◽  
pp. 555-577 ◽  
Author(s):  
D. Cohen ◽  
J. Crampton ◽  
A. Gagarin ◽  
G. Gutin ◽  
M. Jones
Keyword(s):  
Constraint Satisfaction Problem ◽  
Practical Interest ◽  
Satisfiability Problem ◽  
Business Rules ◽  
Generic Algorithm ◽  
Fixed Parameter Tractable ◽  
Novel Approach ◽  
Constraint Language ◽  
Fixed Parameter ◽  
Language User

The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan - an assignment of tasks to authorized users - such that all constraints are satisfied. It is natural to see the WSP as a subclass of the Constraint Satisfaction Problem (CSP) in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive is that the number of tasks is usually very small compared to the number of users, so it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. This novel approach to the WSP, using techniques from CSP, has enabled us to design a generic algorithm which is FPT for several families of workflow constraints considered in the literature. Furthermore, we prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. Lastly, we identify a new FPT constraint language, user-independent constraints, that includes many of the constraints of interest in business processing systems. We demonstrate that our generic algorithm has provably optimal running time O*(2^(klog k)), for this language, where k is the number of tasks.

A Multi-Agent Temporal Constraint Satisfaction System Based on Allen's Interval Algebra and Probabilities

2011 ◽  
pp. 56-76
Author(s):  
Elhadi Shakshuki ◽  
André Trudel ◽  
Yiqing Xu
Keyword(s):  
Distributed System ◽  
Constraint Satisfaction ◽  
Real World ◽  
Constraint Satisfaction Problem ◽  
Local Constraints ◽  
Temporal Interval ◽  
Interval Algebra ◽  
Multi Agent ◽  
The Given ◽  
Execution Order

Many real-world problems can be viewed and represented as a constraint satisfaction problem (CSP). In addition, many of these problems are distributed in nature. To this end, we combine agents with a special type of CSP called an Interval Algebra network (IA network). An IA network is a graph where each node represents an interval. Directed edges in the network are labelled with temporal interval relations. A probabilistic IA network has probabilities associated with the relations on the edges that can be used to capture preferences. A probabilistic IA agent (PIA-Agent) is assigned a probabilistic IA network. PIA-Agent’s networks are connected via edges. The overall goal is to make each PIA-Agent’s network consistent and optimal. Each PIA-Agent is independent and has sole control over its network. But, it must communicate and coordinate with other PIA-Agents when modifying or updating edges that are shared between two PIA-Agents. We present an algorithm which allows the PIA-Agents to collaboratively solve and recommend a temporal schedule. At the agent level, this schedule is optimal under the given local constraints. Although the global solution may not be optimal, we try to generate near optimal ones. Note that our distributed system is not centrally controlled. Our algorithm decides which PIA-Agent should be given an opportunity to update the solution next. Also, when a conflict is detected, the algorithm modifies the PIA-Agent execution order in order to deal with the inconsistency.

scholarly journals THE APPROACH TO MEASUREMENT OF REQUIREMENT QUALITY BY APPLICATION OF GENERALIZED PRIORITIZED FUZZY CONSTRAINT SATISFACTION PROBLEM

2021 ◽  
Vol 19 (3) ◽  
pp. 175
Author(s):  
Radomir Prodanović ◽  
Dejan Rančić ◽  
Ivan Vulić ◽  
Dušan Bogićević
Keyword(s):  
Constraint Satisfaction ◽  
Quality Evaluation ◽  
Constraint Satisfaction Problem ◽  
Quality Measurement ◽  
Development Lifecycle ◽  
Software Development Lifecycle ◽  
Fuzzy Constraint ◽  
Fuzzy Constraint Satisfaction ◽  
The Given

The requirement quality affects product development at all lifecycle stages, as well as the end product. Poorly defined requirements bring to extended deadlines, increased financial costs, even to project disruption. Current researches related to the good quality of requirements include characteristics of good requirements and the development of new elicitation techniques. Requirement quality evaluation should be tailored both to the professionals and users who defined requirements according to their needs. Therefore, the model is designed for requirement quality measurement based on the characteristics of good requirements by application of the Generalized Prioritized Fuzzy Constraint Satisfaction Problem. The model enables the participation of selected characteristics of good requirements in quality evaluation, according to priorities. The evaluator obtains information if the requirement satisfies the given quality satisfaction threshold based on the degree of fulfillment of selected characteristics of a good requirement. The model is applied to all types of requirements, as well as to the evaluation of requirements at all software development lifecycle stages.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals Permutation groups with small orbit growth

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manuel Bodirsky ◽  
Bertalan Bodor
Keyword(s):  
Automorphism Group ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Permutation Groups ◽  
First Order ◽  
Finite Covers

Abstract Let K exp + \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c ⁢ n d ⁢ n cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c , d c,d with d < 1 d<1 . We show that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K exp + \mathcal{K}_{{\operatorname{exp}}{+}} . We also show that Thomas’ conjecture holds for K exp + \mathcal{K}_{{\operatorname{exp}}{+}} : all structures in K exp + \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

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