COCOON 2017: Computing and Combinatorics pp 13-24 | Cite as

An FPTAS for the Volume of Some \(\mathcal{V}\)-polytopes—It is Hard to Compute the Volume of the Intersection of Two Cross-Polytopes

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10392)

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 \((n/\log n)^n\). 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 \(\mathcal{V}\)-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., \(L_1\)-balls) is #P-hard, unlike the cases of \(L_{\infty }\)-balls or \(L_2\)-balls.

Keywords

#P-hard Deterministic approximation FPTAS \(\mathcal{V}\)-polytope Intersection of \(L_1\)-balls 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

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) and by JST PRESTO Grant Number JPMJPR16E4, Japan.

References

  1. 1.
    Ando, E., Kijima, S.: An FPTAS for the volume computation of 0–1 knapsack polytopes based on approximate convolution. Algorithmica 76(4), 1245–1263 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bandyopadhyay, A., Gamarnik, D.: Counting without sampling: asymptotics of the log-partition function for certain statistical physics models. Random Struct. Algorithms 33, 452–479 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bárány, I., Füredi, Z.: Computing the volume is difficult. Discrete Comput. Geom. 2, 319–326 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proceedings of STOC 2007, pp. 122–127 (2007)Google Scholar
  5. 5.
    Cousins, B., Vempala, S., Bypassing, K.L.S.: Gaussian cooling and an \(O^\ast (n^3)\) volume algorithm. In: Proceedings of STOC 2015, pp. 539–548 (2015)Google Scholar
  6. 6.
    Dadush, D., Vempala, S.: Near-optimal deterministic algorithms for volume computation via M-ellipsoids. Proc. Natl. Acad. Sci. USA 110(48), 19237–19245 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dyer, M.: Approximate counting by dynamic programming. In: Proceedings of STOC 2003, pp. 693–699 (2003)Google Scholar
  8. 8.
    Dyer, M., Frieze, A.: On the complexity of computing the volume of a polyhedron. SIAM J. Comput. 17(5), 967–974 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dyer, M., Frieze, A., Kannan, R.: A random polynomial-time algorithm for approximating the volume of convex bodies. J. Assoc. Comput. Mach. 38(1), 1–17 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Elekes, G.: A geometric inequality and the complexity of computing volume. Discrete Comput. Geom. 1, 289–292 (1986)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gamarnik, D., Katz, D.: Correlation decay and deterministic FPTAS for counting list-colorings of a graph. In: Proceedings of SODA 2007, pp. 1245–1254 (2007)Google Scholar
  12. 12.
    Gopalan, P., Klivans, A., Meka, R.: Polynomial-time approximation schemes for knapsack and related counting problems using branching programs. arXiv:1008.3187v1 (2010)
  13. 13.
    Gopalan, P., Klivans, A., Meka, R., Štefankovič, D., Vempala, S., Vigoda, E.: An FPTAS for #knapsack and related counting problems. In: Proceedings of FOCS 2011, pp. 817–826 (2011)Google Scholar
  14. 14.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  15. 15.
    Khachiyan, L.: The problem of computing the volume of polytopes is \(\#P\)-hard. Uspekhi Mat. Nauk. 44, 199–200 (1989)MATHGoogle Scholar
  16. 16.
    Khachiyan, L.: Complexity of polytope volume computation. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry, pp. 91–101. Springer, Berlin (1993)CrossRefGoogle Scholar
  17. 17.
    Li, L., Lu, P., Yin, Y.: Approximate counting via correlation decay in spin systems. In: Proceedings of SODA 2012, pp. 922–940 (2012)Google Scholar
  18. 18.
    Li, L., Lu, P., Yin, Y.: Correlation decay up to uniqueness in spin systems. In: Proceedings of SODA 2013, pp. 67–84 (2013)Google Scholar
  19. 19.
    Li, J., Shi, T.: A fully polynomial-time approximation scheme for approximating a sum of random variables. Oper. Res. Lett. 42, 197–202 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lin, C., Liu, J., Lu, P.: A simple FPTAS for counting edge covers. In: Proceedings of SODA 2014, pp. 341–348 (2014)Google Scholar
  21. 21.
    Lovász, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. Applied Mathematics. SIAM Society for Industrial, Philadelphia (1986)CrossRefMATHGoogle Scholar
  22. 22.
    Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an \(O^\ast (n^4)\) volume algorithm. J. Comput. Syst. Sci. 72, 392–417 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Matous̆ek, J.: Lectures on Discrete Geometry. Springer, New York (2002)CrossRefGoogle Scholar
  24. 24.
    Štefankovič, D., Vempala, S., Vigoda, E.: A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM J. Comput. 41(2), 356–366 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Weitz, D.: Counting independent sets up to the tree threshold. In: Proceedings of STOC 2006, pp. 140–149 (2006)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sojo UniversityKumamotoJapan
  2. 2.Kyushu UniversityFukuokaJapan
  3. 3.JST PRESTOFukuokaJapan

Personalised recommendations