Computing the volume is difficult

  title={Computing the volume is difficult},
  author={Imre B{\'a}r{\'a}ny and Zolt{\'a}n F{\"u}redi},
  journal={Discrete \& Computational Geometry},
AbstractFor every polynomial time algorithm which gives an upper bound % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVC0xe9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpi0x% c9q8qqaqFj0df9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaie% aacaWF2bGaa83Baiaa-Xgaaaaaaa!3A2B! $$\overline {vol}$$ (K) and a lower boundvol(K) for the volume of a convex setK⊂Rd, the… 

Approximating the volume of convex bodies

AbstractIt is a well-known fact that for every polynomial-time algorithm which gives an upper bound % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn%

Linear Equations Modulo 2 and the L1 Diameter of Convex Bodies

The algorithm is based on a reduction to the problem of computing the diameter of a convex body in Rn with respect to the L1 norm and it is shown that it is possible to do so up to a multiplicative error of O(radic(n/logn)), while no randomized polynomial time algorithm can achieve accuracy O( Radic(logn) O(1).

An FPTAS for the Volume of a V-polytope - It is Hard to Compute The Volume of The Intersection of Two Cross-polytopes

This paper investigates the volume of a "knapsack dual polytope," which is known to be #P-hard due to Khachiyan (1989), and reduces an approximate volume to that of the intersection of two cross-polytopes, and gives FPTASs for those volume computations.

Polynomial time algorithms to approximate mixed volumes within a simply exponential factor

  • L. Gurvits
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2007
The mixed volume analogues of the Van der Waerden and the Schrijver/Valiant conjectures on the permanent are proved and ”justify” the convex relaxation, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.

An FPTAS for the Volume Computationof 0-1 Knapsack Polytopes Based on Approximate Convolution Integral

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 is presented.

On the Complexity of Computing Mixed Volumes

A randomized algorithm based on polynomial-time randomized algorithms for computing the volume of convex bodies is derived and applications are presented to various problems in discrete mathematics, combinatorics, computational convexity, algebraic geometry, geometry of numbers, and operations research.

Quantum algorithm for estimating volumes of convex bodies

A quantum algorithm that estimates the volume of an $n$-dimensional convex body within multiplicative error $\epsilon$ using $\tilde{O}(n^{3}+n^{2.5}/\ep silon)$ queries to a membership oracle and additional arithmetic operations, which is the first quantum speedup for volume estimation.

Parallel degree computation for solution space of binomial systems with an application to the master space of $\mathcal{N}=1$ gauge theories

A specialized parallel algorithm for computing the degree on GPUs that takes advantage of the massively parallel nature of GPU devices and achieves nearly 30 fold speedup over its CPU-only counterpart enabling the discovery of previously unknown results.

The Distribution Function of the Longest Path Length in Constant Treewidth DAGs with Random Edge Length

  • E. Ando
  • Computer Science, Mathematics
  • 2019
It is shown that there is a fully polynomial time approximation scheme (FPTAS) for computing $\Pr[X_{\rm MAX}\le x]$ in case the treewidth of $G$ is bounded by a constant $k$, where there may be exponentially many $s-t$ paths in $G$.



A geometric inequality and the complexity of computing volume

  • G. Elekes
  • Mathematics
    Discret. Comput. Geom.
  • 1986
The volume of the convex hull of anym points of ann-dimensional ball with volumeV is at mostV·m/2n. This implies that no polynomial time algorithm can compute the volume of a convex set (given by an

Algorithmic theory of numbers, graphs and convexity

  • L. Lovász
  • Mathematics
    CBMS-NSF regional conference series in applied mathematics
  • 1986
How to Round Numbers Preliminaries and some Applications in Combinatorics Cuts and Joins Chromatic Number, Cliques and Perfect Graphs Minimizing a Submodular Function.

An extremal property of the hypersphere

  • A. Macbeath
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1951
It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its

Stochastical approximation of convex bodies

Enveloppe convexe de points aleatoires sur la sphere. Enveloppe convexe de points aleatoires sur la frontiere d'un corps convexe lisse


Philosophical and ideological approach to the study of childhood as a social phenomenon is substantiated in the article. The meaning of human existence and his relationship with the surrounding

An algorithmic theory of numbers, graphs and convexity, Preprint

  • Report No. 85368-OR,
  • 1985

Sections euclidiennes et volume des corps symetriques convexes darts R

  • C. R. Acad. Sci. Paris Sdr. I
  • 1985

Approximation ofthe ball by polytopes having few vertices

    Sections euclidiennes et volume des corps symetriques convexes dans

    • C. R. Acad. Sci. Paris Sár
    • 1985