# A random polynomial-time algorithm for approximating the volume of convex bodies

@article{Dyer1989ARP, title={A random polynomial-time algorithm for approximating the volume of convex bodies}, author={Martin E. Dyer and Alan M. Frieze and Ravi Kannan}, journal={J. ACM}, year={1989}, volume={38}, pages={1-17} }

A randomized polynomial-time algorithm for approximating the volume of a convex body <italic>K</italic> in <italic>n</italic>-dimensional Euclidean space is presented. The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within <italic>K</italic>.

## 763 Citations

### A New Approach to Polynomial-Time Generation of Random Points in Convex Bodies

- Mathematics
- 1996

In this paper we describe a new method for proving the polynomial-time convergence of an algorithm for sampling (almost) uniformly at random from a convex body in high dimension. Previous approaches…

### An elementary analysis of a procedure for sampling points in a convex body

- MathematicsRandom Struct. Algorithms
- 1998

A new method for proving the convergence of an algorithm for sampling almost uniformly at random from a convex body in high dimension by using a more elementary coupling argument.

### Computing the volume of convex bodies : a case where randomness provably helps

- Computer Science, Mathematics
- 1991

The problem of computing the volume of a convex body K in R is discussed and worst-case results are reviewed and randomised approximation algorithms which show that with randomisation one can approximate very nicely are provided.

### A Polynomial Number of Random Points Does Not Determine the Volume of a Convex Body

- MathematicsDiscret. Comput. Geom.
- 2011

We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in ℝn, can approximate the volume of the body up to a constant factor…

### Random walks and an O*(n5) volume algorithm for convex bodies

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 1997

This work introduces three new ideas: the use of the isotropic position (or at least an approximation of it) for rounding, the separation of global obstructions and local obstructions for fast mixing, and a stepwise interlacing of rounding and sampling.

### ALGORITHM FOR CONVEX BODIES

- Mathematics, Computer Science
- 1999

This work introduces three new ideas: the use of the isotropic position (or at least an approximation of it) for rounding, the separation of global obstructions and local obstructions for fast mixing, and a stepwise interlacing of rounding and sampling.

### Approximation of diameters: randomization doesn't help

- MathematicsProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

We describe a deterministic polynomial-time algorithm which, for a convex body K in Euclidean n-space, finds upper and lower bounds on K's diameter which differ by a factor of O(/spl radic/n/logn).…

### Random Walks in a Convex Body and an Improved Volume Algorithm

- MathematicsRandom Struct. Algorithms
- 1993

A randomized algorithm using O(n7 log’ n) separation calls to approximate the volume of a convex body with a fixed relative error is given and the mixing rate of Markov chains from finite to arbitrary Markov Chains is analyzed.

### Approximating the Volume of General Pfaffian Bodies

- Mathematics, Computer ScienceStructures in Logic and Computer Science
- 1997

We introduce a new powerful method of approximating the volume (and integrals) of vast number of geometric bodies defined by boolean combinations of Pfaffian conditions. The method depends on the…

### Deterministic and randomized polynomial-time approximation of radii

- Mathematics
- 2001

This paper is concerned with convex bodies in n-dimensional l p spaces, where each body is accessible only by a weak separation or optimization oracle. It studies the asymptotic relative accuracy, as…

## References

SHOWING 1-10 OF 21 REFERENCES

### A geometric inequality and the complexity of computing volume

- MathematicsDiscret. 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

- MathematicsCBMS-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.

### On the Complexity of Computing the Volume of a Polyhedron

- Computer ScienceSIAM J. Comput.
- 1988

We show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix.

### Monte-Carlo algorithms for enumeration and reliability problems

- Mathematics24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
- 1983

A simple but very general Monte-Carlo technique for the approximate solution of enumeration and reliability problems and several applications are given.

### Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains

- MathematicsWG
- 1987

The general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods: this in turn leads to randomised approximate counting algorithms for these graphs with very good asymptotic behaviour.

### Geometric Algorithms and Combinatorial Optimization

- MathematicsAlgorithms and Combinatorics
- 1988

0. Mathematical Preliminaries.- 0.1 Linear Algebra and Linear Programming.- Basic Notation.- Hulls, Independence, Dimension.- Eigenvalues, Positive Definite Matrices.- Vector Norms, Balls.- Matrix…

### How hard is it to marry at random? (On the approximation of the permanent)

- Computer ScienceSTOC '86
- 1986

Al though finding a perfect matching is easy and finding a Hamil tonian circuit is hard, counting perfect matchings and counting Hamiltonian circuits is equally hard, as hard as computing the number of solutions of any problem in NP.

### Asymptotic Theory Of Finite Dimensional Normed Spaces

- Mathematics
- 1986

The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.-…

### Random walks on finite groups and rapidly mixing markov chains

- Mathematics
- 1983

© Springer-Verlag, Berlin Heidelberg New York, 1983, tous droits reserves. L’acces aux archives du seminaire de probabilites (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml),…

### An Introduction to Probability Theory and Its Applications

- Mathematics
- 1950

Thank you for reading an introduction to probability theory and its applications vol 2. As you may know, people have look numerous times for their favorite novels like this an introduction to…