## Abstract

The Densest κ-Subgraph (DκS) problem, and its corresponding minimization problem Smallest p-Edge Subgraph (SpES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both DkS and SpES from graphs to hypergraphs. We consider the Densest k-Subhypergraph problem (given a hypergraph (V,E), find a subset W C V of k vertices so as to maximize the number of hyperedges contained in W) and define the Minimum p-Union problem (given a hypergraph, choose p of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n^{4} (4-√^{3})/^{13+}ϵ) ≤ O(n^{0.697831+ϵ})-approximation (for arbitrary constant ϵ > 0) for Densest κ-Subhypergraph and an O(√m)-approximation for Minimum p-Union. We also give an O(p m)-Approximation for Minimum p-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016 |

Editors | Klaus Jansen, Claire Mathieu, Jose D. P. Rolim, Chris Umans |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770187 |

DOIs | |

State | Published - 1 Sep 2016 |

Externally published | Yes |

Event | 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 - Paris, France Duration: 7 Sep 2016 → 9 Sep 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 60 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 |
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Country/Territory | France |

City | Paris |

Period | 7/09/16 → 9/09/16 |

### Bibliographical note

Funding Information:Partially supported by NSF grants 1464239 and 1535887.

## Keywords

- Approximation algorithms
- Hypergraphs