Recent Results on the Algebraic Approach to the CSP

Bulatov, Andreǐ A.; Valeriote, Matthew

doi:10.1007/978-3-540-92800-3_4

Cited by 57 publications

(59 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is the basis for the success of the algebraic approach to the CSP. We refer to [8,11] for an introduction to the algebraic approach and to [6] for the connection to positive primitive interpretability. Here we only define algebraic notions required for the proof of the main result.…”

Section: Characterization Of Bounded Widthmentioning

confidence: 99%

The collapse of the bounded width hierarchy

Barto

2014

J Logic Computation

147

View full text Add to dashboard Cite

show abstract

Section: Characterization Of Bounded Widthmentioning

confidence: 99%

The collapse of the bounded width hierarchy

Barto

2014

J Logic Computation

147

View full text Add to dashboard Cite

show abstract

“…This conjecture, which is a strengthening of the dichotomy conjecture, has come to be known as the algebraic dichotomy conjecture. There are several equivalent formulations of this conjecture (for additional information, see for example [16]). We state a version here which was realized through algebraic work due to Maróti and McKenzie [77].…”

Section: The Algebraic Approachmentioning

confidence: 99%

On Some Combinatorial Optimization Problems : Algorithms and Complexity

Uppman¹

2015

View full text Add to dashboard Cite

This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems.A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs.The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case.Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest.In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of iv optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.This work has been supported in part by the National Graduate School in Computer Science (CUGS), Sweden. Populärvetenskaplig sammanfattningOptimering går ut på att hitta värden för ett antal variabler så att resultatet blir så bra som möjligt (enligt en given kostnadsfunktion). Vi studerar kombinatoriska optimeringsproblem, problem där man för varje variabel enbart har ett ändligt antal möjliga val. Sådana problem är i någon mening lätta; för att hitta en så bra lösning som möjligt kan man helt enkelt prova alla sätt att välja värden för variablerna. Tyvärr är det här inte någon metod som fungerar i praktiken...

show abstract

“…See [Burris and Sankappanavar 1981] for more detail on basic notions of universal algebra and [Bulatov et al 2005;Bulatov and Valeriote 2008;Cohen and Jeavons 2006] for more detail about the algebraic approach to the CSP.…”

Section: Algebramentioning

confidence: 99%

Robust Satisfiability for CSPs

Dalmau

Krokhin

2013

ACM Trans. Comput. Theory

View full text Add to dashboard Cite

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least a (1−f (ϵ))-fraction of constraints for each (1−ϵ)-satisfiable instance (i.e. such that at most a ϵ-fraction of constraints needs to be removed to make the instance satisfiable), where f (ϵ) → 0 as ϵ → 0. We establish an algebraic framework for analyzing constraint satisfaction problems admitting an efficient robust algorithm with functions f of a given growth rate. We use this framework to derive hardness results. We also describe three classes of problems admitting an efficient robust algorithm such that f is O(1/ log (1/ϵ)), O(ϵ 1/k ) for some k > 1, and O(ϵ), respectively. Finally, we give a complete classification of robust satisfiability with a given f for the Boolean case.

show abstract

Recent Results on the Algebraic Approach to the CSP

Abstract: Abstract. We describe an algebraic approach to the constraint satisfaction problem (CSP) and present recent results on the CSP that make use of, in an essential way, this algebraic framework.

Cited by 57 publications

References 30 publications

The collapse of the bounded width hierarchy

The collapse of the bounded width hierarchy

On Some Combinatorial Optimization Problems : Algorithms and Complexity

Robust Satisfiability for CSPs

Contact Info

Product

Resources

About