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

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

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Article ID: iaor20023429
Country: United Kingdom
Volume: 29
Issue: 5
Start Page Number: 441
End Page Number: 458
Publication Date: Apr 2002
Journal: Computers and Operations Research
Authors: , , , ,
Keywords: graphs, heuristics
Abstract:

Let G = (V, E) be a connected graph with specified vertex υ0V, length l(e) ⩾ 0 for each e ∈ 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.

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