Abstract
В работе изучается взаимосвязь между расстоянием Громова — Хаусдорфа и задачами дискретной оптимизации. Расстояние Громова — Хаусдорфа до метрического пространства с одинаковыми непутевыми расстояниями используется используется для решения следующих проблем: вычисление длин ребер минимального остовного дерева для конечного метрического пространства; обобщенная пробам Борсука; вычисление хроматическогочисла и минимального размера клинкового покрытия для простого графа.
Highlights
The aim of the paper is to demonstrate close connections between the geometry of Gromov– Hausdorff distance and such popular Discrete Optimisation problems as minimum spanning tree problem, Borsuk conjecture, estimation of chromatic number and clique cover number of a simple graph
Hausdorff [4] as the infimum of positive numbers r such that A is contained in the r-neighbourhood of B, and vice-versa. It is well-known that this function, referred as the Hausdorff distance, is a metric on the family of all closed bounded subsets of the metric space X, see for example [3]
Ψ Y → Z into all possible metric spaces Z. This value is referred as the Gromov–Hausdorff distance between X and Y. It is well-known that this distance function a metric on the family of isometry classes of compact metric spaces
Summary
The aim of the paper is to demonstrate close connections between the geometry of Gromov– Hausdorff distance and such popular Discrete Optimisation problems as minimum spanning tree problem, Borsuk conjecture, estimation of chromatic number and clique cover number of a simple graph. For a finite subset M of a metric space X, consider the complete graph K(M ) with the vertex set M , endowed with the weight function whose value on an edge {x, y} equals to the distance |xy| between the points x and y in the space X. In the present paper we consider a generalized Borsuk problem, passing to an arbitrary bounded metric space X and its partitions of an arbitrary cardinality m (not necessary finite). A solution to the Borsuk problem for a 2-distance space X with distances a < b is obtained in terms of the clique cover number of the simple graph G with vertex set X, whose vertices x and y are connected by an edge iff |xy| = a. We are infinitely thankful for his deep influence, kind care, permanent support and attention
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