scholarly journals The Tractability of CSP Classes Defined by Forbidden Patterns

2012 ◽  
Vol 45 ◽  
pp. 47-78 ◽  
Author(s):  
D. A. Cohen ◽  
M. C. Cooper ◽  
P. Creed ◽  
D. Marx ◽  
A. Z. Salamon
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
The Novel ◽  
Considerable Effort ◽  
Relational Properties ◽  
Np Complete ◽  
Novel Concept ◽  
Forbidden Patterns ◽  
Special Case ◽  
Hybrid Properties

The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NP-hard in general, considerable effort has been spent on identifying tractable subclasses. The main two approaches consider structural properties (restrictions on the hypergraph of constraint scopes) and relational properties (restrictions on the language of constraint relations). Recently, some authors have considered hybrid properties that restrict the constraint hypergraph and the relations simultaneously. Our key contribution is the novel concept of a CSP pattern and classes of problems defined by forbidden patterns (which can be viewed as forbidding generic sub-problems). We describe the theoretical framework which can be used to reason about classes of problems defined by forbidden patterns. We show that this framework generalises certain known hybrid tractable classes. Although we are not close to obtaining a complete characterisation concerning the tractability of general forbidden patterns, we prove a dichotomy in a special case: classes of problems that arise when we can only forbid binary negative patterns (generic sub-problems in which only disallowed tuples are specified). In this case we show that all (finite sets of) forbidden patterns define either polynomial-time solvable or NP-complete classes of instances.

scholarly journals CSP DICHOTOMY FOR SPECIAL POLYADS

2013 ◽  
Vol 23 (05) ◽  
pp. 1151-1174 ◽  
Author(s):  
LIBOR BARTO ◽  
JAKUB BULÍN
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Np Complete ◽  
Bounded Width ◽  
Near Unanimity

For a digraph ℍ, the Constraint Satisfaction Problem with template ℍ, or CSP(ℍ), is the problem of deciding whether a given input digraph 𝔾 admits a homomorphism to ℍ. The CSP dichotomy conjecture of Feder and Vardi states that for any digraph ℍ, CSP(ℍ) is either in P or NP-complete. Barto, Kozik, Maróti and Niven (Proc. Amer. Math. Soc.137 (2009) 2921–2934) confirmed the conjecture for a class of oriented trees called special triads. We generalize this result, establishing the dichotomy for a class of oriented trees which we call special polyads. We prove that every tractable special polyad has bounded width and provide the description of special polyads of width 1. We also construct a tractable special polyad which neither has width 1 nor admits any near-unanimity polymorphism.

scholarly journals Variable Elimination in Binary CSPs

2019 ◽  
Vol 66 ◽  
pp. 589-624
Author(s):  
Martin C. Cooper ◽  
Achref El Mouelhi ◽  
Cyril Terrioux
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Benchmark Problems ◽  
Study Variable ◽  
Variable Elimination ◽  
Problem Instances ◽  
Forbidden Patterns

We investigate rules which allow variable elimination in binary CSP (constraint satisfaction problem) instances while conserving satisfiability. We study variable-elimination rules based on the language of forbidden patterns enriched with counting and quantification over variables and values. We propose new rules and compare them, both theoretically and experimentally. We give optimised algorithms to apply these rules and show that each define a novel tractable class. Using our variable-elimination rules in preprocessing allowed us to solve more benchmark problems than without.

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 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 Galois Connections for Patterns: An Algebra of Labelled Graphs

2021 ◽  
pp. 125-150
Author(s):  
David A. Cohen ◽  
Martin C. Cooper ◽  
Peter G. Jeavons ◽  
Stanislav Živný
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Galois Connection ◽  
Language Classes ◽  
Finite Set ◽  
Binary Constraint ◽  
Tree Width ◽  
To Come ◽  
Forbidden Patterns ◽  
Near Unanimity

AbstractA pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any $$k \ge 1$$ k ≥ 1 , instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of instances such that each instance has a weak near-unanimity polymorphism of arity k (thus covering all tractable language classes).We investigate the power of forbidding augmented patterns and discuss their potential for describing new tractable classes.

scholarly journals A heuristic approach to constraint optimization in timetabling

2002 ◽  
Vol 20 (1) ◽  
pp. 64 ◽  
Author(s):  
A. Chand
Keyword(s):  
Constraint Satisfaction ◽  
Query Language ◽  
Constraint Satisfaction Problem ◽  
Heuristic Method ◽  
Constraint Optimization ◽  
Examination Timetabling ◽  
Acceptable Solution ◽  
Timetabling Problem ◽  
Complete Problem ◽  
Np Complete

Timetabling is a difficult (NP-complete) problem and belongs to a general class of problems known as scheduling. Due to a variety of constraints typical in different timetabling environments, it has been difficult to develop a generic solution for timetabling. This paper is an attempt to define a generic computational model for examination timetabling for predefined constraints found in the problem, and proposes a heuristic method of developing an acceptable solution. The declarative nature of the developed constraints language (based on the structured query language) is utilized to construct constraints and specify the timetabling problem as a constraint satisfaction problem. A university examination timetabling problem is used to illustrate and test the model.

scholarly journals Statistical physics of hard optimization problems

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
Lenka Zdeborová
Keyword(s):  
Cost Function ◽  
Constraint Satisfaction ◽  
Disordered Systems ◽  
Statistical Physics ◽  
Optimization Problems ◽  
Constraint Satisfaction Problem ◽  
Constraint Satisfaction Problems ◽  
Cavity Method ◽  
Complete Class ◽  
Np Complete

Statistical physics of hard optimization problemsOptimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.

scholarly journals A Recommender System for Mobility-as-a-Service Plans Selection

2021 ◽  
Vol 13 (15) ◽  
pp. 8245
Author(s):  
Konstantina Arnaoutaki ◽  
Efthimios Bothos ◽  
Babis Magoutas ◽  
Attila Aba ◽  
Domokos Esztergár-Kiss ◽  
...  
Keyword(s):  
Recommender System ◽  
Constraint Satisfaction ◽  
Smart Cities ◽  
Constraint Satisfaction Problem ◽  
Real Life ◽  
End Users ◽  
Cosine Similarity ◽  
Experimental Results ◽  
The Real ◽  
Novel Concept

Transportation and mobility in smart cities are undergoing a grave transformation as new ways of mobility are introduced to facilitate seamless traveling, addressing travelers’ needs in a personalized manner. A novel concept that has been recently introduced is Mobility-as-a-Service (MaaS), where mobility services are bundled in MaaS Plans and offered to end-users through a single digital platform. The present paper introduces a recommender system for MaaS Plans selection that supports travelers to select bundles of mobility services that fit their everyday transportation needs. The recommender filters out unsuitable plans and then ranks the remaining ones on the basis of their similarity to the users’ characteristics, habits and preferences. The recommendation approach is based on Constraint Satisfaction Problem (CSP) formalisms combined with cosine similarity techniques. The proposed method was evaluated in experimental settings and was further embedded in real-life pilot MaaS applications. The experimental results showed that the proposed approach provides lists of MaaS PlanMaaS Plans that users would choose in a real-life MaaS setting, in most of the cases. Moreover, the results of the real-life pilots showed that the majority of the participants chose an actual MaaS Plan from the top three places of the recommendation lists.

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