### Abstract

Computing high dimensional volumes is a hard problem, even for approximation. It is known that no polynomial-time deterministic algorithm can approximate with ratio 1.999n the volumes of convex bodies in the n dimension as given by membership oracles. Several randomized approximation techniques for #P-hard problems has been developed in the three decades, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. For instance, Stefankovic, Vempala and Vigoda (2012) gave an FPTAS for counting 0-1 knapsack solutions (i.e., integer points in a 0-1 knapsack polytope) based on an ingenious dynamic programming. Motivated by a new technique for designing FPTAS for #P-hard problems, this paper is concerned with the volume computation of 0-1 knapsack polytopes: it is given by {x (Formula presented.)} with a positive integer vector a and a positive integer b as an input, the volume computation of which is known to be #P-hard. Li and Shi (2014) gave an FPTAS for the problem by modifying the dynamic programming for counting solutions. This paper presents a new technique based on approximate convolution integral for a deterministic approximation of volume computations, and provides an FPTAS for the volume computation of 0-1 knapsack polytopes.

Original language | English |
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Title of host publication | Algorithms and Computation - 25th International Symposium, ISAAC 2014, Proceedings |

Editors | Hee-Kap Ahn, Chan-Su Shin |

Publisher | Springer Verlag |

Pages | 376-386 |

Number of pages | 11 |

ISBN (Electronic) | 9783319130743 |

DOIs | |

Publication status | Published - Jan 1 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8889 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Algorithms and Computation - 25th International Symposium, ISAAC 2014, Proceedings*(pp. 376-386). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8889). Springer Verlag. https://doi.org/10.1007/978-3-319-13075-0_30