A Graph-Theoretic Characterization of the $\text{PV}_{\text{chunk}}$ Class of Synchronizing Primitives

Henderson, Peter B.; Zalcstein, Yechezkel

doi:10.1137/0206008

Cited by 99 publications

(37 citation statements)

References 6 publications

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“…Graphic halfspace functions are Boolean functions associated with threshold graphs, a well-studied class of graphs [24], [28] with applications to synchronizing parallel processes [37].…”

Section: Definitionsmentioning

confidence: 99%

Complexity theoretic hardness results for query learning

et al. 1998

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We investigate the complexity of learning for the well-studied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may make membership queries. We develop a general technique for proving hardness results for learning with membership and equivalence queries (and for more general query models). We apply the technique to show that, assuming NP = co-NP, no polynomial-time membership and (proper) equivalence query algorithms exist for exactly learning read-thrice DNF formulas, unions of k ≥ 3 halfspaces over the Boolean domain, or some other related classes. Our hardness results are representation dependent, and do not preclude the existence of representation independent algorithms.The general technique introduces the representation problem for a class F of representations (e.g., formulas), which is naturally associated with the learning problem for F . This problem is related to the structural question of how to characterize functions representable by formulas in F , and is a generalization of standard complexity problems such as Satisfiability. While in general the representation problem is in Σ P 2 , we present a theorem demonstrating that for "reasonable" classes F , the existence of a polynomial-time membership and equivalence query algorithm for exactly learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard.

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“…Graphic halfspace functions are Boolean functions associated with threshold graphs, a well-studied class of graphs [24], [28] with applications to synchronizing parallel processes [37].…”

Section: Definitionsmentioning

confidence: 99%

Complexity theoretic hardness results for query learning

et al. 1998

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“…A graph is inter& (Chapter 8 of [6]) if there is for each vertex A of G an interval (aA, bJ of the real line, such that two vertices A and B of G are connected by an edge if and only if the corresponding intervals intersect in the real line. A graph G is threshold (Chapter 10 of [6], also [l] and [7]) if there exists a way of labelling each vertex A of G with a nonnegative integer f(A) and there is another nonnegative integer f (the threshold) such that a set of vertices of G induces at least one edge if and only if the sum of their Iabels exceeds t.…”

Section: Definitionsmentioning

confidence: 99%

The size of chordal, interval and threshold subgraphs

et al. 1989

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“…An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices.…”

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confidence: 99%

Efficient Computation of the Characteristic Polynomial of a Threshold Graph

Fürer

2015

Frontiers in Algorithmics

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Abstract. An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of O(n log 2 n) for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time.

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A Graph-Theoretic Characterization of the $\text{PV}_{\text{chunk}}$ Class of Synchronizing Primitives

Cited by 99 publications

References 6 publications

Complexity theoretic hardness results for query learning

Complexity theoretic hardness results for query learning

The size of chordal, interval and threshold subgraphs

Efficient Computation of the Characteristic Polynomial of a Threshold Graph

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