Max-Sur-CSP on Two Elements

Uppman, Hannes

doi:10.1007/978-3-642-33558-7_6

Cited by 3 publications

(5 citation statements)

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“…An early result on this problem family was the complexity classification of all two-element structures [7, Proposition 6.11], [8, Proposition 4.7]. There is recent interest in understanding the complexity of these problems, which perhaps focuses on the cases where the structure B is a graph; we refer the reader to the survey [3] for further information and pointers, and also can reference the related articles [13,10,11]. The introduction in the survey [3] suggests that the problems SCSP(B) "seem to be very difficult to classify in terms of complexity", and that "standard methods to prove easiness or hardness fail."…”

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confidence: 99%

An algebraic hardness criterion for surjective constraint satisfaction

Chen

2014

Algebra Univers.

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The constraint satisfaction problem (CSP) on a relational structure B is to decide, given a set of constraints on variables where the relations come from B, whether or not there is a assignment to the variables satisfying all of the constraints; the surjective CSP is the variant where one decides the existence of a surjective satisfying assignment onto the universe of B. We present an algebraic condition on the polymorphism clone of B and prove that it is sufficient for the hardness of the surjective CSP on a finite structure B, in the sense that this problem admits a reduction from a certain fixed-structure CSP. To our knowledge, this is the first result that allows one to use algebraic information from a relational structure B to infer information on the complexity hardness of surjective constraint satisfaction on B. A corollary of our result is that, on any finite non-trivial structure having only essentially unary polymorphisms, surjective constraint satisfaction is NP-complete.

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confidence: 99%

An algebraic hardness criterion for surjective constraint satisfaction

Chen

2014

Algebra Univers.

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“…The generalised Min-Cut problem consists in minimising an objective function f + g, where f is a superadditive set function given by an oracle and g is a cut function (same as in the Min-Cut problem); see Section 5 for the details. We prove that the running time of our algorithm is roughly O n 20α , thus improving on the bound of O n 3 3α established in [43] (one of the two extended conference abstracts of this paper) for the special case of {0, 1}-valued languages.…”

Section: Contributionsmentioning

confidence: 55%

“…In [23], a weighted relation γ is called PDS if both Feas(γ) and Opt(γ) are essentially downsets. For a {0, 1}-valued weighted relation, this condition is equivalent to that of being almost-min-min [43]. By Corollary 30, PDS and EDS are equivalent concepts for languages of finite size.…”

Section: Lemma 25 a Crisp Weighted Relation Is Eds If And Only If Itmentioning

confidence: 98%

“…We would like to thank Yuni Iwamasa, who prompted us to extend the complexity classification to languages of infinite size (and bounded arity). We also thank the anonymous reviewers of the two extended abstracts [43,23] of this paper.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…Finally, we define the Generalised Min-Cut problem, which further generalises the problem introduced in [43].…”

Section: Generalised Min-cut Problemmentioning

confidence: 99%

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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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An Algebraic Hardness Criterion for Surjective Constraint Satisfaction

Chen

2014

Preprint

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Max-Sur-CSP on Two Elements

Cited by 3 publications

References 23 publications

An algebraic hardness criterion for surjective constraint satisfaction

An algebraic hardness criterion for surjective constraint satisfaction

The Complexity of Boolean Surjective General-Valued CSPs

An Algebraic Hardness Criterion for Surjective Constraint Satisfaction

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