The Complexity of Three-Element Min-Sol and Conservative Min-Cost-Hom

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP'13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding valued CSPs is preserved

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“…The following corollary has been observed in the context of Min-Sol-Hom and Min-Cost-Hom [31] by Hannes Uppman. 4 Corollary 39.…”

Section: Lemma 35mentioning

confidence: 67%

A Galois Connection for Weighted (Relational) Clones of Infinite Size

Fulla

Živný

2016

ACM Trans. Comput. Theory

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“…As our main result in this section, we give a complexity classification of all Min-Sol languages on arbitrary finite domains, thus improving on previous classifications obtained for Min-Sol languages on domains with two elements [42], three elements [68], and other special cases [41,40,39].…”

Section: Complexity Consequencesmentioning

confidence: 85%

The Power of Sherali--Adams Relaxations for General-Valued CSPs

Thapper¹,

Živný²

2017

SIAM J. Comput.

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We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems to optimality. The condition is that of bounded width which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs.Our characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many known valued constraint languages are tractable via the third level of the Sherali-Adams relaxation. For the known languages, this is a significantly simpler algorithm than the previously obtained ones. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov andŽivný [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13]. Union is not liable for any use that may be made of the information contained therein.unless we settle the CSP dichotomy conjecture [30], some additional requirement on the unary weighted relations (such as injectivity) is necessary.As a corollary of our characterisation, we give, in Section 3.4, a complete complexity classification of exact solvability of Minimum-Solution problems over arbitrary finite domains, thus improving on previous partial classifications for domains of size two [42] and three [68], homogeneous and maximal (under a certain algebraic conjecture) languages [39] and on graphs with few vertices [41]. Theorem 3.19 shows that the Minimum-Solution problem is NP-hard unless it satisfies the bounded width condition. Previous partial results included ad-hoc algorithms for various special cases. Our result shows that one algorithm, the third level of the Sherali-Adams relaxation, solves all tractable cases and is thus universal. As a matter of fact, we actually prove a complexity classification for a larger class of problems that includes Minimum-Solutions problems as a special case, as described in detail in Section 3.4. Related workThe first level of the Sherali-Adams hierarchy is known as the basic linear programming (BLP) relaxation [16]. In [63], the authors gave a precise algebraic characterisation of Γ for which any instance of VCSP(Γ) is solved to optimality by BLP, see also [44]. The characterisation proved important not only in the study of VCSPs [36] and other classes of problems [34], but also in the design of fixed-parameter algorithms [37]. In [66], it was then shown that for finite-valued CSPs, the BLP solves all tractable cases; i.e. if BLP fails to solve any instance of some finite-valued constraint language then this la...

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“…Min-Sols generalise Min-Ones [14] and bounded integer linear programs. Min-Sols have been only very recently classified [47] with respect to computational complexity, thus improving on previous partial classifications [27][28][29][30]48]. Minimum Cost Homomorphism (Min-Cost-Hom) problems are Valued CSPs in which all but unary weighted relations are {0, ∞}-valued.…”

Section: Introductionmentioning

confidence: 99%

“…Min-Cost-Hom problems with all unary cost functions have been classified in [43]. Also, Min-Cost-Hom problems with all unary {0, ∞}-valued cost functions [44,48] and on three-element domains [49] have been classified.…”

Section: Introductionmentioning

confidence: 99%

Necessary Conditions for Tractability of Valued CSPs

Thapper¹,

Živný²

2015

SIAM J. Discrete Math.

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The connection between constraint languages and clone theory has been a fruitful line of research on the complexity of constraint satisfaction problems. In a recent result, Cohen et al. [SICOMP'13] have characterised a Galois connection between valued constraint languages and so-called weighted clones. In this paper, we study the structure of weighted clones. We extend the results of Creed andŽivný from [CP'11/SICOMP'13] on types of weightings necessarily contained in every nontrivial weighted clone. This result has immediate computational complexity consequences as it provides necessary conditions for tractability of weighted clones and thus valued constraint languages. We demonstrate that some of the necessary conditions are also sufficient for tractability, while others are provably not.

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The Complexity of Three-Element Min-Sol and Conservative Min-Cost-Hom

Cited by 14 publications

References 25 publications

A Galois Connection for Weighted (Relational) Clones of Infinite Size

A Galois Connection for Weighted (Relational) Clones of Infinite Size

The Power of Sherali--Adams Relaxations for General-Valued CSPs

Necessary Conditions for Tractability of Valued CSPs

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