## Abstract

We discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player.

We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS.

We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS.

Original language | English |
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Title of host publication | Algorithmic Decision Theory |

Subtitle of host publication | 5th International Conference, ADT 2017, Luxembourg, Luxembourg, October 25–27, 2017, Proceedings |

Editors | Jorg Rothe |

Place of Publication | Dordrecht |

Publisher | Springer |

Pages | 228-240 |

Number of pages | 13 |

ISBN (Electronic) | 978-3-319-67504-6 |

ISBN (Print) | 978-3-319-67503-9 |

DOIs | |

Publication status | Published - 2017 |

Event | 5th International Conference on Algorithmic Decision Theory (ADT 2017) - Luxembourg, Luxembourg Duration: 25 Oct 2017 → 27 Oct 2017 Conference number: 5 https://sma.uni.lu/adt2017/ |

### Publication series

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

Volume | 10576 |

### Conference

Conference | 5th International Conference on Algorithmic Decision Theory (ADT 2017) |
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Abbreviated title | ADT 2017 |

Country | Luxembourg |

City | Luxembourg |

Period | 25/10/17 → 27/10/17 |

Internet address |