### Abstract

The input to the min sum set cover problem is a collection of n sets that jointly cover m elements. The output is a linear order on the sets, namely, in every time step from 1 to n exactly one set is chosen. For every element, this induces a first time step by which it is covered. The objective is to find a linear arrangement of the sets that minimizes the sum of these first time steps over all elements. We show that a greedy algorithm approximates min sum set cover within a ratio of 4. This result was implicit in work of Bar-Noy, Bellare, Halldorsson, Shachnai, and Tamir (1998) on chromatic sums, but we present a simpler proof. We also show that for every ε > 0, achieving an approximation ratio of 4 - ε is NP-hard. For the min sum vertex cover version of the problem (which comes up as a heuristic for speeding up solvers of semidefinite programs) we show that it can be approximated within a ratio of 2, and is NP-hard to approximate within some constant ρ > 1.

Original language | English |
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Pages (from-to) | 219-234 |

Number of pages | 16 |

Journal | Algorithmica (New York) |

Volume | 40 |

Issue number | 4 |

DOIs | |

Publication status | Published - Sep 1 2004 |

### Keywords

- Greedy algorithm
- NP-hardness
- Randomized rounding
- Threshhold

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

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## Cite this

*Algorithmica (New York)*,

*40*(4), 219-234. https://doi.org/10.1007/s00453-004-1110-5