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An FPTAS for the Volume Computation of 0-1 Knapsack Polytopes Based on Approximate Convolution

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Abstract

Computing high dimensional volumes is a hard problem, even for approximation. Several randomized approximation techniques for #P-hard problems have been developed in the three decades, while some deterministic approximation algorithms are recently developed only for a few #P-hard problems. Motivated by a new technique for a deterministic approximation, this paper is concerned with the volume computation of 0-1 knapsack polytopes, which is known to be #P-hard. This paper presents a new technique based on approximate convolutions for a deterministic approximation of volume computations, and provides a fully polynomial-time approximation scheme for the volume computation of 0-1 knapsack polytopes. We also give an extension of the result to multi-constrained knapsack polytopes with a constant number of constraints.

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Notes

  1. In fact, M is a scaling parameter of our approximation algorithm described in Sect. 3 in detail, that is \(M = 2 \lceil n^2\epsilon ^{-1} \rceil \) for an approximate ratio \(\epsilon \) (\(0 < \epsilon \le 1\)). Arguments in this section also hold even in case that M is real. It might be convenient for this section to assume \(M=1\).

  2. For the simplicity of a description, here we assume that our input is given by rationals \(\tilde{a}_j = (a_j/b)M\) (\(j=1,\ldots ,n\)), since \(G_j\) is defined for \(\tilde{a}_j\) [see (5) and (6)]. It is also noted that M is a parameter of the algorithm (see Lemma 3.4, appearing later).

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Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments. This work is partly supported by Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC)” (Nos. 24106008, 24106005).

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Authors and Affiliations

  1. Sojo University, Kumamoto, Japan

    Ei Ando

  2. Kyushu University, Fukuoka, Japan

    Shuji Kijima

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  1. Ei Ando
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  2. Shuji Kijima
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Correspondence to Ei Ando.

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A preliminary version appeared in [1].

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Ando, E., Kijima, S. An FPTAS for the Volume Computation of 0-1 Knapsack Polytopes Based on Approximate Convolution. Algorithmica 76, 1245–1263 (2016). https://doi.org/10.1007/s00453-015-0096-5

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