### Abstract

Given an n-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while only a few #P-hard problems are known to yield deterministic approximation. Motivated by the problem of deterministically approximating the volume of a -polytope, that is a polytope with a small number of vertices and (possibly) exponentially many facets, this paper investigates the problem of computing the volume of a “knapsack dual polytope,” which is known to be #P-hard due to Khachiyan (1989). We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes, and give FPTASs for those volume computations. Interestingly, computing the volume of the intersection of two cross-polytopes (i.e.,-balls) is #P-hard, unlike the cases of-balls or -balls.

Original language | English |
---|---|

Title of host publication | Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings |

Editors | Yixin Cao, Jianer Chen |

Publisher | Springer Verlag |

Pages | 13-24 |

Number of pages | 12 |

ISBN (Print) | 9783319623887 |

DOIs | |

Publication status | Published - Jan 1 2017 |

Event | 23rd International Conference on Computing and Combinatorics, COCOON 2017 - Hong Kong, China Duration: Aug 3 2017 → Aug 5 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 10392 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 23rd International Conference on Computing and Combinatorics, COCOON 2017 |
---|---|

Country | China |

City | Hong Kong |

Period | 8/3/17 → 8/5/17 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Computing and Combinatorics - 23rd International Conference, COCOON 2017, Proceedings*(pp. 13-24). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10392 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-62389-4_2