# Computing the volume is difficult

@article{Brny1986ComputingTV, title={Computing the volume is difficult}, author={Imre B{\'a}r{\'a}ny and Zolt{\'a}n F{\"u}redi}, journal={Discrete \& Computational Geometry}, year={1986}, volume={2}, pages={319-326} }

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…

## 178 Citations

### Approximating the volume of convex bodies

- MathematicsDiscret. Comput. Geom.
- 1993

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

- Mathematics, Computer ScienceFOCS 2007
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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

- Mathematics, Computer ScienceCOCOON
- 2017

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

- MathematicsElectron. 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 of some V-polytopes - It is hard to compute the volume of the intersection of two cross-polytopes

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2020

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

- Mathematics, Computer ScienceISAAC
- 2014

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

- Computer Science, MathematicsSIAM J. Comput.
- 1998

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

- Computer Science, MathematicsArXiv
- 2019

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

- Computer Science, MathematicsArXiv
- 2015

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

- Computer Science, MathematicsArXiv
- 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 $G$.

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