## Abstract

We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f : 2^{N} → R^{+}, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by Unconstrained Submodular Maximization include Max-Cut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige, Mirrokni, and Vondrák [SIAM J. Comput., 40 (2011), pp. 1133-1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.

Original language | English |
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Pages (from-to) | 1384-1402 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 44 |

Issue number | 5 |

DOIs | |

State | Published - 2015 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015 Society for Industrial and Applied Mathematics.

## Keywords

- Approximation algorithms
- Linear time
- Submodular functions