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 \(1.999^n\) 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 \(\{{\varvec{x}} \in \mathbb {R}^n \mid {\varvec{a}}^{\top } {\varvec{x}} \le b,\ 0 \le x_i \le 1\ (i=1,\ldots ,n)\}\) with a positive integer vector \({\varvec{a}}\) 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.
This work is partly supported by Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC)” (No. 24106008, 24106005).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bandyopadhyay, A., Gamarnik, D.: Counting without sampling: asymptotics of the log-partition function for certain statistical physics models. Random Structures and Algorithms 33, 452–479 (2008)
Bárány, I., Füredi, Z.: Computing the volume is difficult. Discrete Computational Geometry 2, 319–326 (1987)
Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proc. of STOC 2007, pp. 122–127 (2007)
Dyer, M.: Approximate counting by dynamic programming. In: Proc. of STOC 2003, pp. 693–699 (2003)
Dyer, M., Frieze, A.: On the complexity of computing the volume of a polyhedron. SIAM Journal on Computing 17(5), 967–974 (1988)
Dyer, M., Frieze, A., Kannan, R.: A random polynomial-time algorithm for approximating the volume of convex bodies. Journal of the Association for Computing Machinery 38(1), 1–17 (1991)
Elekes, G.: A geometric inequality and the complexity of computing volume. Discrete Computational Geometry 1, 289–292 (1986)
Gamarnik, D., Katz, D.: Correlation decay and deterministic FPTAS for counting list-colorings of a graph. In: Proc. of SODA 2007, pp. 1245–1254 (2007)
Gopalan, P., Klivans, A., Meka, R.: Polynomial-time approximation schemes for knapsack and related counting problems using branching programs, arXiv:1008.3187v1 (2010)
Gopalan, P., Klivans, A., Meka, R., Štefankovič, D., Vempala, S., Vigoda, E.: An FPTAS for #knapsack and related counting problems. In: Proc. of FOCS 2011, pp. 817–826 (2011)
Ko, K.-I.: Complexity Theory of Real Functions. Birkhäuser, Boston (1991)
Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proc. of SODA 2012, pp. 922–940 (2012)
Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proc. of SODA 2013, pp. 67–84 (2013)
Li, J., Shi, T.: A fully polynomial-time approximation scheme for approximating a sum of random variables. Operations Research Letters 42, 197–202 (2014)
Lin, C., Liu, J., Lu, P.: A simple FPTAS for counting edge covers. In: Proc. of SODA 2014, pp. 341–348 (2014)
Lovász, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM Society for industrial and applied mathematics, Philadelphia (1986)
Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an \(O^\ast (n^4)\) volume algorithm. Journal of Computer and System Sciences 72, 392–417 (2006)
Mitra, S.: On the probability distribution of the sum of uniformly distributed random variables. SIAM Journal on Applied Mathematics 20(2), 195–198 (1971)
Štefankovič, D., Vempala, S., Vigoda, E.: A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM Journal on Computing 41(2), 356–366 (2012)
Weihrauch, K.: Computable Analysis An Introduction. Springer, Berlin (2000)
Weitz, D.: Counting independent sets up to the tree threshold. In: Proc. STOC 2006, pp. 140–149 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Ando, E., Kijima, S. (2014). An FPTAS for the Volume Computationof 0-1 Knapsack Polytopes Based on Approximate Convolution Integral. In: Ahn, HK., Shin, CS. (eds) Algorithms and Computation. ISAAC 2014. Lecture Notes in Computer Science(), vol 8889. Springer, Cham. https://doi.org/10.1007/978-3-319-13075-0_30
Download citation
DOI: https://doi.org/10.1007/978-3-319-13075-0_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-13074-3
Online ISBN: 978-3-319-13075-0
eBook Packages: Computer ScienceComputer Science (R0)