Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems

Baker, Kirby A.; Pixley, Alden F.

doi:10.1007/bf01187059

Cited by 205 publications

(163 citation statements)

References 13 publications

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“…Lin, McKee, and West [26] proved that for every chordal graph H with n vertices, the leafage is at most n − lg n − 1 2 lg lg n + O (1). Hence, each chordal graph with n vertices admits a NUF of arity n − lg n − 1 2 lg lg n + O(1).…”

Section: Arity Bounds Based On Leafagementioning

confidence: 99%

“…. , p t , u 1 Notice that, in this path, all coordinates with the exception of coordinate k − 1 alternate between u i and v i . We will describe this situation by saying that coordinate k − 1 traverses that path P 0,1 while all other coordinates alternate.…”

Section: T B)mentioning

confidence: 99%

“…, x k ) = a whenever at least k − 1 of the x i 's equal a. Early papers on near-unanimity functions include [1,20]. Near-unanimity functions of arity 3, also called majority functions, are much studied [2,17,31].…”

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confidence: 99%

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Near-Unanimity Functions and Varieties of Reflexive Graphs

Brewster¹,

Feder²,

Hell³

et al. 2008

SIAM J. Discrete Math.

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Abstract. Let H be a graph and k ≥ 3. A near-unanimity function of arity k is a mapping g from the k-tuples over V (H) to V (H) such that g(x 1 , x 2 , . . . , x k ) is adjacent to g(x 1 , x 2 , . . . , x k ) whenever x i x i ∈ E(H) for each i = 1, 2, . . . , k, and g(x 1 , x 2 , . . . , x k ) = a whenever at least k − 1 of the x i 's equal a. Feder and Vardi proved that, if a graph H admits a near-unanimity function, then the homomorphism extension (or retraction) problem for H is polynomial time solvable. We focus on near-unanimity functions on reflexive graphs. The best understood are reflexive chordal graphs H: they always admit a near-unanimity function. We bound the arity of these functions in several ways related to the size of the largest clique and the leafage of H, and we show that these bounds are tight. In particular, it will follow that the arity is bounded by n − √ n + 1, where n = |V (H)|. We investigate substructures forbidden for reflexive graphs that admit a near-unanimity function. It will follow, for instance, that no reflexive cycle of length at least four admits a near-unanimity function of any arity. However, we exhibit nonchordal graphs which do admit near-unanimity functions. Finally, we characterize graphs which admit a conservative near-unanimity function. This characterization has been predicted by the results of Feder, Hell, and Huang. Specifically, those results imply that, if P = NP, the graphs with conservative near-unanimity functions are precisely the so-called bi-arc graphs. We give a proof of this statement without assuming P = NP.

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Section: Arity Bounds Based On Leafagementioning

confidence: 99%

Section: T B)mentioning

confidence: 99%

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Near-Unanimity Functions and Varieties of Reflexive Graphs

Brewster¹,

Feder²,

Hell³

et al. 2008

SIAM J. Discrete Math.

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“…Using a theorem of Baker and Pixley from [4] it follows that if A is a finite algebra that has a k-ary near unanimity term operation (see Example 4) then the function s A (n) is bounded above by a polynomial of degree k − 1 and so such algebras have few subpowers. An early tractability result of Jeavons, Cohen and Cooper [30] establishes that algebras having near unanimity terms are tractable, and it is no coincidence that this tractability result can be proved using the Baker-Pixley theorem.…”

Section: Definition 5 ([2])mentioning

confidence: 99%

Recent Results on the Algebraic Approach to the CSP

Bulatov

Valeriote

2008

Complexity of Constraints

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“…This gives an implicit description of the term clone by infinitely many relations. For certain algebraic structures, like finite lattices, already finitely many relations determine the clone [3]. Hence we say that finite lattices are finitely related (or have finite degree).…”

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confidence: 99%

On finitely related semigroups

Mayr

2012

Semigroup Forum

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An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid.

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Polynomial interpolation and the Chinese Remainder Theorem for algebraic systems

Cited by 205 publications

References 13 publications

Near-Unanimity Functions and Varieties of Reflexive Graphs

Near-Unanimity Functions and Varieties of Reflexive Graphs

Recent Results on the Algebraic Approach to the CSP

On finitely related semigroups

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