We investigate the problem of the most efficient first-order definition of the property of containing an induced subgraph isomorphic to a given pattern graph, which is closely related to the time complexity of the decision problem for this property.
We derive a series of new bounds for the minimum quantifier depth of a formula defining this property for pattern graphs on five vertices, as well as for disjoint unions of isomorphic complete multipartite graphs. Moreover, we prove that for any
there exists a graph on
vertices and a first-order formula of quantifier depth at most
that defines the property of containing an induced subgraph isomorphic to this graph.
Bibliography: 12 titles.
В настоящей работе изучается задача наиболее эффективной записи на языке первого порядка свойства наличия индуцированного подграфа, изоморфного заданному pattern-графу, тесно связанная с оцениванием временно́й сложности проверки такого свойства. Мы получили ряд новых оценок наименьшей кванторной глубины формулы, определяющей упомянутое свойство для pattern-графов на пяти вершинах, а также для дизъюнктных объединений изоморфных полных многодольных графов. Кроме того, мы доказали, что для любого $\ell\geq 4$ найдется граф на $\ell$ вершинах и формула первого порядка кванторной глубины не более $\ell-1$, записывающая свойство содержать индуцированный подграф, изоморфный этому графу.
Библиография: 12 названий.
Locality-sensitive hashing [IM98] is a classical data structure for approximate nearest neighbor search. It allows, after a close to linear time preprocessing of the input dataset, to find an approximately nearest neighbor of any fixed query in sublinear time in the dataset size. The resulting data structure is randomized and succeeds with high probability for every fixed query. In many modern applications of nearest neighbor search the queries are chosen adaptively. In this paper, we study the robustness of the locality-sensitive hashing to adaptive queries in Hamming space. We present a simple adversary that can, under mild assumptions on the initial point set, provably find a query to the approximate near neighbor search data structure that the data structure fails on. Crucially, our adaptive algorithm finds the hard query exponentially faster than random sampling.
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