## Abstract

In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the running time of O(

We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.

*n*log*n*) , where*n*is the number of items. Before, only a 3-approximation algorithm was known.We note that our problem with a convex cost function is a special case of maximizing a non-monotone, possibly negative submodular function.

Original language | English |
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Title of host publication | Mathematical Foundations of Computer Science 2013 |

Subtitle of host publication | 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings |

Editors | Krishnendu Chatterjee, Jirí Sgall |

Place of Publication | Berlin, Heidelberg |

Publisher | Springer |

Pages | 62-73 |

ISBN (Electronic) | 978-3-642-40313-2 |

ISBN (Print) | 978-3-642-40312-5 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

Event | 38th International Symposium on Mathematical Foundations of Computer Science, MFCS 2013 - Klosterneuburg, Austria Duration: 26 Aug 2013 → 30 Aug 2013 Conference number: 38 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 8087 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 38th International Symposium on Mathematical Foundations of Computer Science, MFCS 2013 |
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Abbreviated title | MFCS |

Country/Territory | Austria |

City | Klosterneuburg |

Period | 26/08/13 → 30/08/13 |