On the complexity of the Cable-Trench Problem
The Cable-Trench Problem (CTP) is a common generalization of the Single-Source Shortest Paths Problem (SSSP) and the Minimum Spanning Tree Problem (MST): given an edge-weighted graph with a special root vertex and parameters τ,γ≥0, the goal is to find a spanning tree that minimizes the total edge co...
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| Veröffentlicht in: | Discrete Applied Mathematics Jg. 340; S. 272 - 285 |
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15.12.2023
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| Abstract | The Cable-Trench Problem (CTP) is a common generalization of the Single-Source Shortest Paths Problem (SSSP) and the Minimum Spanning Tree Problem (MST): given an edge-weighted graph with a special root vertex and parameters τ,γ≥0, the goal is to find a spanning tree that minimizes the total edge costs plus the total cost of the paths from each vertex to the root, scaled by τ and γ, respectively. While it is well known that both SSSP and MST can be solved in polynomial time, CTP is NP-hard. We show that computing an approximate solution with factor less than 1.000475 is NP-hard, thus ruling out a polynomial-time approximation scheme, unless P=NP.
We also consider the more general Steiner Cable-Trench Problem (SCTP), for which only a given subset of terminal vertices must be spanned by a solution. The tree might include non-terminal vertices, known as Steiner vertices, although only paths from terminals to the root are considered in the total cost. For this problem, we present a (2.88+ϵ)-approximation based on a counting argument, for any ϵ>0; also, we give a simple parameterized algorithm with the number of terminals as parameter. |
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| AbstractList | The Cable-Trench Problem (CTP) is a common generalization of the Single-Source Shortest Paths Problem (SSSP) and the Minimum Spanning Tree Problem (MST): given an edge-weighted graph with a special root vertex and parameters τ,γ≥0, the goal is to find a spanning tree that minimizes the total edge costs plus the total cost of the paths from each vertex to the root, scaled by τ and γ, respectively. While it is well known that both SSSP and MST can be solved in polynomial time, CTP is NP-hard. We show that computing an approximate solution with factor less than 1.000475 is NP-hard, thus ruling out a polynomial-time approximation scheme, unless P=NP.
We also consider the more general Steiner Cable-Trench Problem (SCTP), for which only a given subset of terminal vertices must be spanned by a solution. The tree might include non-terminal vertices, known as Steiner vertices, although only paths from terminals to the root are considered in the total cost. For this problem, we present a (2.88+ϵ)-approximation based on a counting argument, for any ϵ>0; also, we give a simple parameterized algorithm with the number of terminals as parameter. |
| Author | Rosado, Hugo K.K. Pedrosa, Lehilton L.C. Benedito, Marcelo P.L. |
| Author_xml | – sequence: 1 givenname: Marcelo P.L. surname: Benedito fullname: Benedito, Marcelo P.L. email: mplb@ic.unicamp.br – sequence: 2 givenname: Lehilton L.C. surname: Pedrosa fullname: Pedrosa, Lehilton L.C. email: lehilton@ic.unicamp.br – sequence: 3 givenname: Hugo K.K. orcidid: 0000-0002-8881-9699 surname: Rosado fullname: Rosado, Hugo K.K. email: hugo.rosado@ic.unicamp.br |
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