Euclidean networks with a backbone and a limit theorem for minimum spanning caterpillars

Petar Jevtić, J. Michael Steele

Research output: Contribution to journalArticle

Abstract

A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of G leaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the "last mile problem." Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals.

Original languageEnglish (US)
Pages (from-to)992-1004
Number of pages13
JournalMathematics of Operations Research
Volume40
Issue number4
DOIs
StatePublished - Nov 2015

Keywords

  • Beardwood Halton and Hammersley theorem
  • Caterpillar graphs
  • Euclidean networks
  • Gutman graphs
  • Minimal spanning trees
  • Shortest paths
  • Subadditive Euclidean functional
  • Traveling salesman problem

ASJC Scopus subject areas

  • Mathematics(all)
  • Computer Science Applications
  • Management Science and Operations Research

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