34,424 research outputs found

    Covering pairs by q2 + q + 1 sets

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    AbstractFor given k and s let n(k, s) be the largest cardinality of a set whose pairs can be covered by sk-sets. We determine n(k, q2 + q + 1) if a PG(2, q) exists, k > q(q + 1)2, and the remainder of k divided by (q + 1) is at least √q. Asymptotic results are also given for n(k, s) whenever s is fixed and k → ∞. Our main tool is the theory of fractional matchings of hypergraphs

    Disproving the normal graph conjecture

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    A graph GG is called normal if there exist two coverings, C\mathbb{C} and S\mathbb{S} of its vertex set such that every member of C\mathbb{C} induces a clique in GG, every member of S\mathbb{S} induces an independent set in GG and C∩S≠∅C \cap S \neq \emptyset for every C∈CC \in \mathbb{C} and S∈SS \in \mathbb{S}. It has been conjectured by De Simone and K\"orner in 1999 that a graph GG is normal if GG does not contain C5C_5, C7C_7 and C7‾\overline{C_7} as an induced subgraph. We disprove this conjecture

    Interactive Submodular Set Cover

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    We introduce a natural generalization of submodular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results.Comment: 15 pages, 1 figur

    Quantum causal histories

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    Quantum causal histories are defined to be causal sets with Hilbert spaces attached to each event and local unitary evolution operators. The reflexivity, antisymmetry, and transitivity properties of a causal set are preserved in the quantum history as conditions on the evolution operators. A quantum causal history in which transitivity holds can be treated as ``directed'' topological quantum field theory. Two examples of such histories are described.Comment: 16 pages, epsfig latex. Some clarifications, minor corrections and references added. Version to appear in Classical and Quantum Gravit
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