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

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>.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[3]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[4]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[5]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[6]  Richard M. Karp,et al.  Monte-Carlo algorithms for enumeration and reliability problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[7]  P. Bérard,et al.  Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov , 1985 .

[8]  Andrei Z. Broder,et al.  How hard is it to marry at random? (On the approximation of the permanent) , 1986, STOC '86.

[9]  Zoltán Füredi,et al.  Computing the volume is difficult , 1986, STOC '86.

[10]  György Elekes,et al.  A geometric inequality and the complexity of computing volume , 1986, Discret. Comput. Geom..

[11]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[12]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[13]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[14]  Mark Jerrum,et al.  Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved , 1988, STOC '88.

[15]  Martin E. Dyer,et al.  On the Complexity of Computing the Volume of a Polyhedron , 1988, SIAM J. Comput..

[16]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[17]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .