The Complexity of Valued Constraint Satisfaction Problems

Živný, Stanislav

doi:10.1007/978-3-642-33974-5

Cited by 41 publications

(38 citation statements)

References 204 publications

(467 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The expressibility of a language of cost functions is precisely characterised by its weighted polymorphisms. The indicator problem for languages of relations (see Section 3) has been generalised to the weighted indicator problem for languages of cost functions [159].…”

Section: Optimisation Versions Of the Cspmentioning

confidence: 99%

Tractability in constraint satisfaction problems: a survey

Carbonnel

Cooper

2015

Constraints

View full text Add to dashboard Cite

Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. What is tractability?The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [83], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [118]. A computational decision problem (such as the constraint satisfaction problem or CSP) consists of a generic instance (in the case of the CSP, a set of variables, their

show abstract

Section: Optimisation Versions Of the Cspmentioning

confidence: 99%

Tractability in constraint satisfaction problems: a survey

Carbonnel

Cooper

2015

Constraints

View full text Add to dashboard Cite

show abstract

“…Additional information about the valued constraint satisfaction problems can be found in [96]. See also [53].…”

Section: Related Areas Of Researchmentioning

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

“…A valued constraint satisfaction problem (VCSP) is a general framework for discrete optimization (see [19] for details). Informally, the VCSP framework deals with the minimization problem of a function represented as the sum of "small" arity functions.…”

Section: Introductionmentioning

confidence: 99%

Discrete convexity in joint winner property

Iwamasa

Murota

Živný

2018

Discrete Optimization

Self Cite

View full text Add to dashboard Cite

In this paper, we reveal a relation between joint winner property (JWP) in the field of valued constraint satisfaction problems (VCSPs) and M ♮ -convexity in the field of discrete convex analysis (DCA). We introduce the M ♮ -convex completion problem, and show that a function f satisfying the JWP is Z-free if and only if a certain function f associated with f is M ♮ -convex completable. This means that if a function is Z-free, then the function can be minimized in polynomial time via M ♮ -convex intersection algorithms. Furthermore we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.• To describe the connection of JWP and M ♮ -convexity, we introduce the M ♮ -convex completion problem, and give a characterization of M ♮ -convex completability.• By utilizing a DCA interpretation of JWP, we propose a new algorithm for Z-free function minimization, which is faster than previous algorithms for some parameter values.This study will hopefully be the first step towards fruitful interactions between VCSPs and DCA.Notations. Let R and R + denote the sets of reals and nonnegative reals, respectively. In this paper, functions can take the infinite value +∞, where a < +∞, a + ∞ = +∞ for a ∈ R, and 0 · (+∞) = 0. Let R := R ∪ {+∞} and R +

show abstract

The Complexity of Valued Constraint Satisfaction Problems

Cited by 41 publications

References 204 publications

Tractability in constraint satisfaction problems: a survey

Tractability in constraint satisfaction problems: a survey

On Some Combinatorial Optimization Problems : Algorithms and Complexity

Discrete convexity in joint winner property

Contact Info

Product

Resources

About