scholarly journals Algebraic Approach to Promise Constraint Satisfaction

2021 ◽  
Vol 68 (4) ◽  
pp. 1-66
Author(s):  
Libor Barto ◽  
Jakub Bulín ◽  
Andrei Krokhin ◽  
Jakub Opršal
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Algebraic Approach ◽  
Graph Colouring ◽  
Finite Domain ◽  
New Class ◽  
The Past ◽  
Open Questions ◽  
Constraint Language ◽  
Special Cases

The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the past 20 years. A new version of the CSP, the promise CSP (PCSP), has recently been proposed, motivated by open questions about the approximability of variants of satisfiability and graph colouring. The PCSP significantly extends the standard decision CSP. The complexity of CSPs with a fixed constraint language on a finite domain has recently been fully classified, greatly guided by the algebraic approach, which uses polymorphisms—high-dimensional symmetries of solution spaces—to analyse the complexity of problems. The corresponding classification for PCSPs is wide open and includes some long-standing open questions, such as the complexity of approximate graph colouring, as special cases. The basic algebraic approach to PCSP was initiated by Brakensiek and Guruswami, and in this article, we significantly extend it and lift it from concrete properties of polymorphisms to their abstract properties. We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem and show that every PCSP with a fixed constraint language is equivalent to a problem of this form. This allows us to identify a “measure of symmetry” that is well suited for comparing and relating the complexity of different PCSPs via the algebraic approach. We demonstrate how our theory can be applied by giving both general and specific hardness/tractability results. Among other things, we improve the state-of-the-art in approximate graph colouring by showing that, for any k ≥ 3, it is NP-hard to find a (2 k -1)-colouring of a given k -colourable graph.

scholarly journals Tractability of quantified temporal constraints to the max

2014 ◽  
Vol 24 (08) ◽  
pp. 1141-1156 ◽  
Author(s):  
Manuel Bodirsky ◽  
Hubie Chen ◽  
Michał Wrona
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Temporal Constraints ◽  
Temporal Constraint ◽  
First Order ◽  
Constraint Language ◽  
Constraint Languages

A temporal constraint language is a set of relations that are first-order definable over (ℚ;<). We show that several temporal constraint languages whose constraint satisfaction problem is maximally tractable are also maximally tractable for the more expressive quantified constraint satisfaction problem. These constraint languages are defined in terms of preservation under certain binary polymorphisms. We also present syntactic characterizations of the relations in these languages.

scholarly journals Supermodularity on chains and complexity of maximum constraint satisfaction

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Vladimir Deineko ◽  
Peter Jonsson ◽  
Mikael Klasson ◽  
Andrei Krokhin
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Finite Domain ◽  
Finite Collection ◽  
Simple Description ◽  
Weighted Constraints ◽  
Finite Set ◽  
International Audience ◽  
Overlapping Sets

International audience In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.

scholarly journals Fine-Grained Time Complexity of Constraint Satisfaction Problems

2021 ◽  
Vol 13 (1) ◽  
pp. 1-32
Author(s):  
Peter Jonsson ◽  
Victor Lagerkvist ◽  
Biman Roy
Keyword(s):  
Constraint Satisfaction ◽  
Time Complexity ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Finite Domain ◽  
Infinite Domain ◽  
Worst Case ◽  
Fine Grained ◽  
Restricted Problems ◽  
Np Complete

We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation S D such that the time complexity of the NP-complete problem CSP({ S D }) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({ S D }) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.

scholarly journals A Variable Depth Search Algorithm for Binary Constraint Satisfaction Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
N. Bouhmala
Keyword(s):  
Constraint Satisfaction ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Wide Spectrum ◽  
Constraint Satisfaction Problem ◽  
Practical Interest ◽  
Constraint Satisfaction Problems ◽  
Variable Depth ◽  
The Past ◽  
Binary Constraint

The constraint satisfaction problem (CSP) is a popular used paradigm to model a wide spectrum of optimization problems in artificial intelligence. This paper presents a fast metaheuristic for solving binary constraint satisfaction problems. The method can be classified as a variable depth search metaheuristic combining a greedy local search using a self-adaptive weighting strategy on the constraint weights. Several metaheuristics have been developed in the past using various penalty weight mechanisms on the constraints. What distinguishes the proposed metaheuristic from those developed in the past is the update ofkvariables during each iteration when moving from one assignment of values to another. The benchmark is based on hard random constraint satisfaction problems enjoying several features that make them of a great theoretical and practical interest. The results show that the proposed metaheuristic is capable of solving hard unsolved problems that still remain a challenge for both complete and incomplete methods. In addition, the proposed metaheuristic is remarkably faster than all existing solvers when tested on previously solved instances. Finally, its distinctive feature contrary to other metaheuristics is the absence of parameter tuning making it highly suitable in practical scenarios.

scholarly journals Tractable Set Constraints

2012 ◽  
Vol 45 ◽  
pp. 731-759 ◽  
Author(s):  
M. Bodirsky ◽  
M. Hils
Keyword(s):  
Artificial Intelligence ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Description Logics ◽  
Constraint Satisfaction Problems ◽  
Algebraic Characterization ◽  
Crucial Importance ◽  
Hard Constraint ◽  
Set Constraints ◽  
Constraint Language

Many fundamental problems in artificial intelligence, knowledge representation, and verification involve reasoning about sets and relations between sets and can be modeled as set constraint satisfaction problems (set CSPs). Such problems are frequently intractable, but there are several important set CSPs that are known to be polynomial-time tractable. We introduce a large class of set CSPs that can be solved in quadratic time. Our class, which we call EI, contains all previously known tractable set CSPs, but also some new ones that are of crucial importance for example in description logics. The class of EI set constraints has an elegant universal-algebraic characterization, which we use to show that every set constraint language that properly contains all EI set constraints already has a finite sublanguage with an NP-hard constraint satisfaction problem.

scholarly journals Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 908
Author(s):  
Perla Celis ◽  
Rolando de la Cruz ◽  
Claudio Fuentes ◽  
Héctor W. Gómez
Keyword(s):  
Survival Data ◽  
Maximum Likelihood Estimates ◽  
New Class ◽  
New Family ◽  
Gamma Distributions ◽  
Special Cases ◽  
Log Normal ◽  
Positive Support ◽  
Positive Distributions ◽  
Family Of Distributions

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution.

scholarly journals A New Class of Estimators Based on a General Relative Loss Function

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1138
Author(s):  
Tao Hu ◽  
Baosheng Liang
Keyword(s):  
Loss Function ◽  
Asymptotically Normal ◽  
New Class ◽  
Statistical Inferences ◽  
Relative Loss ◽  
Special Cases ◽  
Class Of Estimators ◽  
Box Cox Transformation ◽  
Loss Estimator ◽  
Prostate Cancer Study

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

scholarly journals Recent progress understanding pathophysiology and genesis of brain AVM—a narrative review

2021 ◽  
Author(s):  
Hans-Jakob Steiger
Keyword(s):  
Large Degree ◽  
Advanced Imaging ◽  
Genetic Origin ◽  
The Past ◽  
Open Questions ◽  
Brain Avm ◽  
Pubmed Search ◽  
Substantial Progress ◽  
The Individual ◽  
Arterial Aneurysms

AbstractConsiderable progress has been made over the past years to better understand the genetic nature and pathophysiology of brain AVM. For the actual review, a PubMed search was carried out regarding the embryology, inflammation, advanced imaging, and fluid dynamical modeling of brain AVM. Whole-genome sequencing clarified the genetic origin of sporadic and familial AVM to a large degree, although some open questions remain. Advanced MRI and DSA techniques allow for better segmentation of feeding arteries, nidus, and draining veins, as well as the deduction of hemodynamic parameters such as flow and pressure in the individual AVM compartments. Nonetheless, complete modeling of the intranidal flow structure by computed fluid dynamics (CFD) is not possible so far. Substantial progress has been made towards understanding the embryology of brain AVM. In contrast to arterial aneurysms, complete modeling of the intranidal flow and a thorough understanding of the mechanical properties of the AVM nidus are still lacking at the present time.

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