Asking the Metaquestions in Constraint Tractability

Chen, Hubie; Larose, Benoît

doi:10.1145/3134757

Cited by 24 publications

(43 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is an open question whether condition (2) in Theorem 5 is decidable. We remark that the decidability question for the related property of having symmetric polymorphisms of all arities (see Theorem 3) is also open -see [50,51] for related results. However, another related property -of having so-called fractional symmetric polymorphisms of all arities -is decidable [42].…”

Section: Theorem 8 ([48]) For Any Integer K ≥ 3 and Any Real Numbermentioning

confidence: 99%

Towards a characterization of constant-factor approximable finite-valued CSPs

Dalmau

Krokhin

Manokaran

2018

Journal of Computer and System Sciences

View full text Add to dashboard Cite

In this paper we study the approximability of (Finite-)Valued Constraint Satisfaction Problems (VCSPs) with a fixed finite constraint language Γ consisting of finitary functions on a fixed finite domain. An instance of VCSP is given by a finite set of variables and a sum of functions belonging to Γ and depending on a subset of the variables. Each function takes values in [0, 1] specifying costs of assignments of labels to its variables, and the goal is to find an assignment of labels to the variables that minimizes the sum. A recent result of Ene et al. says that, under the mild technical condition that Γ contains the function corresponding to the equality relation, the basic LP relaxation is optimal for constant-factor approximation for VCSP(Γ) unless the Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give new natural algebraic conditions for the finiteness of the integrality gap for the basic LP relaxation of VCSP(Γ). We also show how these algebraic conditions can in principle be used to round solutions of the basic LP relaxation, and how this leads to efficient constant-factor approximation algorithms for several examples that cover all previously known cases that are NP-hard to solve to optimality but admit constant-factor approximation. Finally, we show that the absence of another algebraic condition leads to NP-hardness of constant-factor approximation. Thus, our results strongly indicate where the boundary of constant-factor approximability for VCSPs lies. ✩ This article is an extended version of a paper published in the proceedings of SODA'15. (Rajsekar Manokaran) 21, 32]). Much work around the Unique Games Conjecture (UGC) directly concerns CSPs [21]. This conjecture states that, for any ǫ > 0, there is a large enough number k = k(ǫ) such that it NP-hard to tell ǫ-satisfiable from (1 − ǫ)-satisfiable instances of CSP(Γ k ), where Γ k consists of all graphs of bijections on a k-element set. Many approximation algorithms for classical optimization problems have been shown optimal assuming the UGC [21,32]. Raghavendra proved [17] that one SDP-based algorithm provides optimal approximation for all problems GCSP(Γ) assuming the UGC. In this paper, we investigate problems VCSP(Γ) and Min CSP(Γ) on an arbitrary finite domain that belong to APX, i.e. admit a (polynomial-time) constant-factor approximation algorithm, proving some results that strongly indicate where the boundary of this property lies.Related Work. Note that each problem Max CSP(Γ) trivially admits a constantfactor approximation algorithm because a random assignment of values to the variables is guaranteed to satisfy a constant fraction of constraints; this can be derandomized by the standard method of conditional probabilities. The same also holds for GCSP. Clearly, for Min CSP(Γ) to admit a constant-factor approximation algorithm, CSP(Γ) must be polynomial-time solvable.The approximability of problems VCSP(Γ) has been studied, mostly for Min CSPs in the Boolean case (i.e., with domain {0, 1}, such CSPs are sometimes ...

show abstract

Section: Theorem 8 ([48]) For Any Integer K ≥ 3 and Any Real Numbermentioning

confidence: 99%

Towards a characterization of constant-factor approximable finite-valued CSPs

Dalmau

Krokhin

Manokaran

2018

Journal of Computer and System Sciences

View full text Add to dashboard Cite

show abstract

“…Bulatov, Jeavons and Krokhin [8] was to show that if a finite relational structure A has no proper retracts and fails a particular special condition on its polymorphisms, then CSP(A) is NP-complete. Using the results of Barto and Kozik [5, Theorem 4.1] and then Chen and Larose [11,Lemma 6.4], the special condition can be stated as: there exists a cyclic polymorphism. For our purposes we will use the following equivalent condition, also from [5, Theorem 4.1]: for all primes p > |A| there is a cyclic polymorphism of A arity p. The authors' All or Nothing Theorem [18,Theorem 5.2] shows that the result of [8] can be transfered to the membership problem for the quasivariety (and universal Horn class) of A: if A has no cyclic polymorphism, then QMEM(A) is NP-complete with respect to first order reductions.…”

Section: Hardness and Nonfinite Axiomatisability A Fundamental Contrmentioning

confidence: 99%

Axiomatisability and hardness for universal Horn classes of hypergraphs

Ham

Jackson

2018

Algebra Univers.

View full text Add to dashboard Cite

We characterise finite axiomatisability and intractability of deciding membership for universal Horn classes generated by finite loop-free hypergraphs.A universal Horn class is a class of model-theoretic structures of the same signature, closed under taking ultraproducts (P u ), direct products over nonempty families (P) and isomorphic copies of substructures (S); see [9,17,30,32] for example. Equivalently they are classes axiomatisable by way of universal Horn sentences: universally quantified disjunctions α 1 ∨ · · · ∨ α k , where each α i is either an atomic formula of the language, or a negated atomic formula, and all but at most one of the α i are negated. Quasivarieties are very closely related classes, differing from the universal Horn class definition only in that the trivial one-element structure (in which all relations are total) is automatically included; this corresponds to allowing the degenerate direct product over an empty family of structures.Problems of axiomatisability for universal Horn classes and quasivarieties have a relatively long history. The starting point is perhaps Maltsev's characterisation of semigroups embeddable in groups [28,29], with subsequent developments in semigroup theory including Sapir [36], Margolis and Sapir [31], Jackson and Volkov [25]. There is also a wealth of literature within universal algebra and relational structures; see Gorbunov's book [17], or the Studia Logica special issue [2] for many examples. An extra impetus for investigation of universal Horn classes comes from computational complexity. For example, the fixed template constraint satisfaction problem over a finite relational structure is the problem of deciding membership of relational structures in a certain universal Horn class [23]. Computational issues for universal Horn classes of relational structures also play a hidden role behind a number of examples demonstrating intractability of deciding membership of finite algebras in a finitely generated pseudovariety. Indeed, several of the relatively few known examples involve encoding a NP-complete universal Horn class membership problem into a pseudovariety membership problems. This is true for Szekely [37], Jackson and McKenzie [24] and [21] for example.The present article concerns both axiomatisability and computational complexity for universal Horn classes of loop-free hypergraphs, and we are able to extend all of the known results for simple graphs. The characterisation of finitely axiomatisable universal Horn classes of finite simple graphs was given by Caicedo [10], by combining a probabilistic result of Erdős [14] with work of Nešetřil and Pultr [34]. In fact, Caicedo's work covers any universal Horn class whose members have bounded chromatic number. After fixing a reasonable model-theoretic meaning to "hypergraph" we show that Caicedo's classification may be extended to arbitrary loop-free

show abstract

“…Given a class T of constraint languages, the meta-problem (or metaquestion [8]) for T takes as input a constraint language Γ and asks if Γ ∈ T . In the context of CSP and c-CSP, the class T is often defined as the set of all languages that admit a combination of polymorphisms satisfying a certain set of identities; in this case the metaproblem is a polymorphism detection problem.…”

Section: Meta-problems and Identitiesmentioning

confidence: 99%

“…We will be interested in particular sets of identities called linear strong Mal'tsev conditions. Given that universal algebra is not the main topic of our paper, we will use a simplified exposition similar to that found in [8]. A linear identity is an expression of the form f (…”

Section: Meta-problems and Identitiesmentioning

confidence: 99%

“…By extension, we say that a constraint language satisfies a linear strong Mal'tsev condition M if it has a collection of polymorphisms that satisfy M. The definability of a class of constraint languages by a linear strong Mal'tsev condition M is strongly tied up with the meta-problem, because for such classes we can associate any constraint language Γ with a polynomial-sized CSP instance whose solutions, if any, are exactly the polymorphisms of Γ satisfying M [8]. We will describe the construction below.…”

Section: Meta-problems and Identitiesmentioning

confidence: 99%

See 1 more Smart Citation

The Dichotomy for Conservative Constraint Satisfaction is Polynomially Decidable

Carbonnel

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. Given a fixed constraint language Γ , the conservative CSP over Γ (denoted by c-CSP(Γ )) is a variant of CSP(Γ ) where the domain of each variable can be restricted arbitrarily. In [5] a dichotomy has been proven for conservative CSP: for every fixed language Γ , c-CSP(Γ ) is either in P or NP-complete. However, the characterization of conservatively tractable languages is of algebraic nature and the recognition algorithm provided in [5] is super-exponential in the domain size. The main contribution of this paper is a polynomial-time algorithm that, given a constraint language Γ as input, decides if c-CSP(Γ ) is tractable. In addition, if Γ is proven tractable the algorithm also outputs its coloured graph, which contains valuable information on the structure of Γ .

show abstract

Asking the Metaquestions in Constraint Tractability

Cited by 24 publications

References 48 publications

Towards a characterization of constant-factor approximable finite-valued CSPs

Towards a characterization of constant-factor approximable finite-valued CSPs

Axiomatisability and hardness for universal Horn classes of hypergraphs

The Dichotomy for Conservative Constraint Satisfaction is Polynomially Decidable

Contact Info

Product

Resources

About