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An Explicit Solution of a Generalized Optimum Requirement Spanning Tree Problem With a Property Related to Monge
Author(s) -
Anazawa Tsutomu
Publication year - 2001
Publication title -
international transactions in operational research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.032
H-Index - 52
eISSN - 1475-3995
pISSN - 0969-6016
DOI - 10.1111/1475-3995.00264
Subject(s) - mathematics , generalization , spanning tree , distributed minimum spanning tree , sort , steiner tree problem , mathematical optimization , minimum spanning tree , tree (set theory) , property (philosophy) , function (biology) , combinatorics , mathematical analysis , arithmetic , evolutionary biology , philosophy , epistemology , biology
The paper considers a generalization of the optimum requirement spanning tree problem (ORST problem) first studied by Hu in 1974. Originally, ORST was regarded as a communication network of tree type with the minimum average cost, and it is obtained by the well‐known Gomory–Hu algorithm when the degrees of vertices are not restricted. The ORST problem is generalized by (i) generalizing the objective function and (ii) imposing maximum degree constraints. The generalized ORST problem includes some practical problems, one of which is proposed in this paper, but is not efficiently solvable in general. However, I show that a particular tree (which is obtained by a sort of greedy algorithm but is explicitly definable) is a solution of the generalized problem when a certain practical condition is satisfied. The condition is closely related to the Monge property, which is originally discussed in the Hitchcock transportation problem, and is known to make some NP‐hard problems efficiently solvable.