The cable trench problem: combining the shortest path and minimum spanning tree problems

  • Authors:

  • Francis J. Vasko;Robert S. Barbieri;Brian Q. Rieksts;Kenneth L. Reitmeyer;Kenneth L. Stott, Jr.

  • Affiliations:

  • Mathematics & CIS Department, Kutztown University, Kutztown, PA, and Homer Research Laboratories, Bethlehem Steel Corporation, Bethlehem, PA;Mathematics & CIS Department, Kutztown University, Kutztown, PA;Mathematics & CIS Department, Kutztown University, Kutztown, PA;Homer Research Laboratories, Bethlehem Steel Corporation, Bethlehem, PA;Homer Research Laboratories, Bethlehem Steel Corporation, Bethlehem, PA

  • Venue:
  • Computers and Operations Research
  • Year:
  • 2002

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Abstract

Let G = (V, E) be a connected graph with specified vertex υ0 &isin V, length l(e) ≥ 0 for each e &isin E, and positive parameters τ and γ. The cable-trench problem (CTP) is to find a spanning tree T such that τlτ(T) + γlγ(T) is minimized where lτ(T) is the total length of the spanning tree T and lγ(T) is the total path length in T from υ0 to all other vertices of V. Since all vertices must be connected to υ0 and only edges from E are allowed, the solution will not be a Steiner tree. Consider the ratio R = τ/γ. For R large enough the solution will be a minimum spanning tree and for R small enough the solution will be a shortest path. In this paper, the CTP will be shown to be NP-complete. A mathematical formulation for the CTP will be provided for specific values of τ and γ. Also, a heuristic will be discussed that will solve the CTP for all values of R.