An algebraic hardness criterion for surjective constraint satisfaction

Chen, Hubie

doi:10.1007/s00012-014-0308-x

Cited by 8 publications

(7 citation statements)

References 15 publications

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“…Firstly, in Section 3, we prove that a reflexive digraph H that is endotrivial has the property that all of its polymorphisms are essentially unary. Combining this result with the aforementioned result of Chen [7] immediately yields that Surjective H-Colouring is NP-complete for any such digraph H. This is the first concrete application of Chen's result to settle a problem of open complexity; it shows, for instance, that Surjective H-Colouring is NP-complete if H is a reflexive directed cycle on k ≥ 3 vertices. As the case k ≤ 2 is trivial, this gives a classification of Surjective H-Colouring for reflexive directed cycles, which we believe form a natural class of digraphs to consider given the results in [24,31].…”

Section: H-retraction H-compaction Surj H-colouring H-colouringsupporting

confidence: 53%

“…Theorem 1 (Corollary 3.5 in [7]). Let H be a finite structure whose universe V (H) has size strictly greater than 1.…”

Section: Essential Unarity and A Dichotomy For Reflexive Directed Cyclesmentioning

confidence: 99%

“…However, the algebraic method is not so far advanced for surjective homomorphism problems. So far it only exists in the work of Chen [7], who proved that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms depend only on one variable, that is, are essentially unary. Chen's result has not yet been put to work (even on toy open problems) and a key driver for our research has been to find, in the wild, a place for its application.…”

Section: H-retraction H-compaction Surj H-colouring H-colouringmentioning

confidence: 99%

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Surjective H-Colouring over Reflexive Digraphs

Larose

Martin

Paulusma

2018

ACM Trans. Comput. Theory

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The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

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Section: H-retraction H-compaction Surj H-colouring H-colouringsupporting

confidence: 53%

“…Theorem 1 (Corollary 3.5 in [7]). Let H be a finite structure whose universe V (H) has size strictly greater than 1.…”

Section: Essential Unarity and A Dichotomy For Reflexive Directed Cyclesmentioning

confidence: 99%

Section: H-retraction H-compaction Surj H-colouring H-colouringmentioning

confidence: 99%

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Surjective H-Colouring over Reflexive Digraphs

Larose

Martin

Paulusma

2018

ACM Trans. Comput. Theory

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“…In the surjective setting, it is the lack of constants that makes it difficult, if not impossible, to employ the algebraic approach. Chen made the first step in this direction [9] but it is not clear how to take his result (for CSPs) further.…”

Section: Related Workmentioning

confidence: 99%

The Complexity of Boolean Surjective General-Valued CSPs

Fulla

Uppman

Živný

2018

ACM Trans. Comput. Theory

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Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q ∪ {∞})-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from D = {0, 1} and an optimal assignment is required to use both labels from D. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory.We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for {0, ∞}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hébrard. For the maximisation problem of Q ≥0 -valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability.Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest. * Extended abstracts of parts of this work appeared in

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“…In fact, there are concrete surjective CSPs defined by a template with only six elements whose complexity is not understood [15] while there are no such open cases for standard CSPs. The complexity of surjective CSPs is subject to significant research activities [15,23] and it appears to be a widely open question whether a dichotomy between PTime and NP holds for the complexity of surjective CSPs. A generalized surjective CSP is defined by a finite set Γ of templates rather than by a single template and the problem is to decide whether there is a surjective homomorphism from the input structure to some interpretation in Γ.…”

Section: Introductionmentioning

confidence: 99%

The Data Complexity of Ontology-Mediated Queries with Closed Predicates

Lutz¹,

Seylan²,

Wolter³

2019

Logical Methods in Computer Science ; Volume 15

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We study the data complexity of ontology-mediated queries in which selected predicates can be closed (OMQCs), carrying out a non-uniform analysis of OMQCs in which the ontology is formulated in one of the lightweight description logics DL-Lite and EL or in the expressive description logic ALCHI. We focus on separating tractable from non-tractable OMQCs. On the level of ontologies, we prove a dichotomy between FO-rewritable and coNP-complete for DL-Lite and between PTime and coNP-complete for EL. We also show that in both cases, the meta problem to decide tractability is in PTime. On the level of OMQCs, we show that there is no dichotomy (unless NP equals PTime) if both concept and role names can be closed. For the case where only concept names can be closed, we tightly link the complexity of OMQC evaluation to the complexity of generalized surjective CSPs. We also identify a useful syntactic class of OMQCs based on DL-LiteR that are guaranteed to be FO-rewritable. LOGICAL METHODS lI N COMPUTER SCIENCE

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An algebraic hardness criterion for surjective constraint satisfaction

Cited by 8 publications

References 15 publications

Surjective H-Colouring over Reflexive Digraphs

Surjective H-Colouring over Reflexive Digraphs

The Complexity of Boolean Surjective General-Valued CSPs

The Data Complexity of Ontology-Mediated Queries with Closed Predicates

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