## Abstract

Given an undirected graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider two network design problems under the power minimization criteria. In both problems we are given a graph G=(V,E) with edge costs and a set T⊆V of terminals. The goal is to find a minimum power edge subset F⊆E such that the graph H=(V,F) satisfies some prescribed requirements. In the MIN-POWER EDGE-COVER problem, H should contain an edge incident to every terminal. Using the Iterative Randomized Rounding (IRR) method, we give an algorithm with expected approximation ratio 1.41; the ratio is reduced to 73/60<1.217 when T is an independent set in G. In the case of unit costs we also achieve ratio 73/60, and in addition give a simple efficient combinatorial algorithm with ratio 5/4. For all these NP-hard problems the previous best known ratio was 3/2. In the related MIN-POWER TERMINAL BACKUP problem, H should contain a path from every t∈T to some node in T∖{t}. We obtain ratio 3/2 for this NP-hard problem, improving the trivial ratio of 2.

Original language | English |
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Pages (from-to) | 285-300 |

Number of pages | 16 |

Journal | Theoretical Computer Science |

Volume | 795 |

DOIs | |

State | Published - 26 Nov 2019 |

### Bibliographical note

Funding Information:Partially supported by NSF grant number 1540547.

Funding Information:

Partially supported by NSF grant number 1540547. Gruia and Zeev thank Neil Olver for many useful discussions. Also, Gruia and Zeev acknowledge the support of the Hausdorff Trimester Program for Combinatorial Optimization (held at the Hausdorff Research Institute for Mathematics, University of Bonn, in 2015). We thank an anonymous reviewer for suggesting the separation-based method for solving the linear program LP(T).

Publisher Copyright:

© 2019 Elsevier B.V.

## Keywords

- Approximation algorithms
- Edge-cover
- Iterative randomized rounding
- Minimum power
- Terminal backup